University of Canberra/RCC2011/RecentChangesCamp Expanded/Sydney

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Date and time: 16 March, 6-8pm, 

Location: Cancer Council Australia

Ground floor, 120 Chalmers St, Boardroom

==[[WikipediaOS|Registration]]==
Please register your attendance on our eventbrite page
http://rccxsydney.eventbrite.com

==Organisers==
* [[user:Chriswaterguy|Chris]] - Wikipedian &amp; co-founder of [http://www.appropedia.org/ Appropedia]
* [[user:Juttavd|Jutta]] Cancer Council Australia - Project Manager - Wiki Development
* Liam Wyatt - Wikimedia Foundation’s Cultural Partnerships Fellow

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Date and time: 16 March, 6-8pm, 

Location: Cancer Council Australia

Ground floor, 120 Chalmers St, Boardroom

==Registration==
Please register your attendance on our eventbrite page
http://rccxsydney.eventbrite.com

==Organisers==
* [[user:Chriswaterguy|Chris]] - Wikipedian &amp; co-founder of [http://www.appropedia.org/ Appropedia]
* [[user:Juttavd|Jutta]] Cancer Council Australia - Project Manager - Wiki Development
* Liam Wyatt - Wikimedia Foundation’s Cultural Partnerships Fellow

{{CourseCat}}</text>
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  <page>
    <title>Universal Bibliography</title>
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{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodicals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

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

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

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

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

Music
*Tele-Tunes

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

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

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

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. [https://books.google.com/books?id=tTzDEAAAQBAJ]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

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

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

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

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

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

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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      <text bytes="55023" sha1="knmpqm6m6z55ve1iq8fcbdidxt1qgrh" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodicals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

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

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

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

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

Music
*Tele-Tunes

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

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

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

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. [https://books.google.com/books?id=tTzDEAAAQBAJ]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

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

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

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

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

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

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
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      <text bytes="55062" sha1="6n38wavpeacj7sj8h5d3ws933o6ylsp" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodicals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

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

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

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

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

Music
*Tele-Tunes

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

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

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

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

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

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

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

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

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

==Notes==
{{Reflist}}
{{subpagesif}}
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      <text bytes="55073" sha1="9hvlwpudrmz9p6tzvr543fqgw5gsgnw" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodicals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

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

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

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

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

Music
*Tele-Tunes

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

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

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

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

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

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

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

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

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

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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      <text bytes="55101" sha1="b3qu4jb3q3t2q4s3mm2529bgdzfyhao" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodicals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

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

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

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

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

Music
*Tele-Tunes

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

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

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

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

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

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

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

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

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

==Notes==
{{Reflist}}
{{subpagesif}}
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      <text bytes="55100" sha1="j6k8wzym60h2837db1t30x1ngkcdxfp" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodicals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

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

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

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

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

Music
*Tele-Tunes

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

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

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

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

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

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

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

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

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

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

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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      <text bytes="55121" sha1="pad085afwg1p93cvo6iuiobxr3w6pux" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

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

This bibliography is arranged as an index of topics.

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

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

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

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

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

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

Particular works:

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

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

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

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

Australia

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

America

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

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

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

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

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

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

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

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

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

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

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

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

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

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

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

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

SF
*Corgi SF Collectors Library

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

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

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

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

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

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

Film
*BFI Companions

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

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

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

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

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

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

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

General series
*Cambridge Studies in Historical Geography

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

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

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

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

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

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

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

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

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

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

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

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

Military
*The Journal of Military History

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

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

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

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

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

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

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

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

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

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

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

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

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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      <text bytes="55205" sha1="4xce4tcp3n4uued5er5hk2qiizkmesf" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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      <text bytes="55770" sha1="m1mpvlcy0itonruzsrw8av5rliezp4l" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
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      <text bytes="55814" sha1="qravxqe7awvf6asb08zrdw47my4suj6" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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      <text bytes="55900" sha1="g7q5pac2d6jk5doc6m335e30rpzyt3o" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Joseph Roggendorf. Studies in Japanese Culture. Sophia University. Tokyo. 1963. [https://books.google.co.uk/books?id=VnBwAAAAMAAJ]
**Joseph Roggendorf. Studies in Japanese Culture: Tradition and Experiment. Sophia University. Tokyo. 2nd Ed. 2nd printing. 1965. [https://books.google.co.uk/books?id=GHJwAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

===Japanese===
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Joseph Roggendorf (ed). Studies in Japanese Culture. Sophia University. Tokyo. 1963. [https://books.google.co.uk/books?id=VnBwAAAAMAAJ]
**Joseph Roggendorf (ed). Studies in Japanese Culture: Tradition and Experiment. Sophia University. Tokyo. 2nd Ed. 2nd printing. 1965. [https://books.google.co.uk/books?id=GHJwAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

===Korean===
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

===Latin American===
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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      <text bytes="43561" sha1="tqkb1a6wp7ji1t8x0phhra48pa8ulqd" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/Languages|Languages]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Cinema|Cinema]]
*[[Universal Bibliography/Culture|Culture]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]]
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]]
*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Toys==
Periodicals
*Games &amp; Toys: The Leading Trade Journal for Home &amp; Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C
*Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. &quot;Harcourt To Buy Journals From Haire Publishing Co&quot; [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29
*[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. &quot;Playthings bought by Geyer-McAllister&quot; [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

====Japan and Korea====
Bibliography
*Bernard S Silberman. Japan and Korea: A Critical Bibliography. University of Arizona Press. 1962. [https://books.google.co.uk/books?id=y6UIAAAAIAAJ]
*Frank J Shulman. Japan and Korea: An Annotated Bibliography of Doctoral Dissertations in Western Languages 1877-1969. American Library Association. 1970. Routledge. 2013. [https://books.google.co.uk/books?id=xs62AQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
      <sha1>0xihtktmy0u6aeo4e5jcws03ciui3lg</sha1>
    </revision>
  </page>
  <page>
    <title>OpenStax Astronomy</title>
    <ns>0</ns>
    <id>194550</id>
    <revision>
      <id>2830501</id>
      <parentid>2762671</parentid>
      <timestamp>2026-09-02T14:20:34Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2830501</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23952" sha1="4zgnfnkg66rgn423409m4zkcxsivcbf" xml:space="preserve">See also [[OpenStax Astronomy 2e]]

See also [[OpenStax]]


==Resources from OpenStax Astronomy also available on Wikiversity==

'''Chapter summaries''' (pdf): [[:File:OpenStax_Astronomy_CH01_ImageSlideshow.pdf|01]] -
[[:File:OpenStax_Astronomy_CH02_ImageSlideshow.pdf|02]] -
[[:File:OpenStax_Astronomy_CH03_ImageSlideshow.pdf|03]] -
[[:File:OpenStax_Astronomy_CH04_ImageSlideshow.pdf|04]] -
[[:File:OpenStax_Astronomy_CH05_ImageSlideshow.pdf|05]] -
[[:File:OpenStax_Astronomy_CH06_ImageSlideshow.pdf|06]] -
[[:File:OpenStax_Astronomy_CH07_ImageSlideshow.pdf|07]] -
[[:File:OpenStax_Astronomy_CH08_ImageSlideshow.pdf|08]] -
[[:File:OpenStax_Astronomy_CH09_ImageSlideshow.pdf|09]] -
[[:File:OpenStax_Astronomy_CH10_ImageSlideshow.pdf|10]] -
[[:File:OpenStax_Astronomy_CH11_ImageSlideshow.pdf|11]] -
[[:File:OpenStax_Astronomy_CH12_ImageSlideshow.pdf|12]] -
[[:File:OpenStax_Astronomy_CH13_ImageSlideshow.pdf|13]] -
[[:File:OpenStax_Astronomy_CH14_ImageSlideshow.pdf|14]] -
[[:File:OpenStax_Astronomy_CH15_ImageSlideshow.pdf|15]] -
[[:File:OpenStax_Astronomy_CH16_ImageSlideshow.pdf|16]] -
[[:File:OpenStax_Astronomy_CH17_ImageSlideshow.pdf|17]] -
[[:File:OpenStax_Astronomy_CH18_ImageSlideshow.pdf|18]] -
[[:File:OpenStax_Astronomy_CH19_ImageSlideshow.pdf|19]] -
[[:File:OpenStax_Astronomy_CH20_ImageSlideshow.pdf|20]] -
[[:File:OpenStax_Astronomy_CH21_ImageSlideshow.pdf|21]] -
[[:File:OpenStax_Astronomy_CH22_ImageSlideshow.pdf|22]] -
[[:File:OpenStax_Astronomy_CH23_ImageSlideshow.pdf|23]] -
[[:File:OpenStax_Astronomy_CH24_ImageSlideshow.pdf|24]] -
[[:File:OpenStax_Astronomy_CH25_ImageSlideshow.pdf|25]] -
[[:File:OpenStax_Astronomy_CH26_ImageSlideshow.pdf|26]] -
[[:File:OpenStax_Astronomy_CH27_ImageSlideshow.pdf|27]] -
[[:File:OpenStax_Astronomy_CH28_ImageSlideshow.pdf|28]] -
[[:File:OpenStax_Astronomy_CH29_ImageSlideshow.pdf|29]] -
[[:File:OpenStax_Astronomy_CH30_ImageSlideshow.pdf|30]]

* Textbook may be viewed [https://upload.wikimedia.org/wikiversity/en/6/6c/Openstax_Astronomy-LR.pdf '''here'''] or at '''[[:File:Openstax Astronomy-LR.pdf]]'''.  Also included in this resource are online versions of the powerpoint slides at '''[[:Category:Openstax_file/Astronomy]]'''.
===OpenStax Astronomy===
*[[:File:Openstax Astronomy-LR.pdf|OpenStax Astronomy]] (also available directly [https://upload.wikimedia.org/wikiversity/en/6/6c/Openstax_Astronomy-LR.pdf '''here'''])

*[http://cnx.org/content/col11992/latest Latest online version]

*[[:Category:Openstax file/Astronomy#PDF versions of author supplied Powerpoints|'''These pdf copies''' of the author-supplied Powerpoints]] might be more convenient to use during a lecture.

==Resources being developed on Wikiversity==
{{under construction}}
[[File:Anonymous_Life_in_the_Universe.pdf|right|250px|thumb|[[:File:Anonymous_Life_in_the_Universe.pdf|You '''must''' read this term paper written by a college freshman.]]]]This is a redacted copy of an email sent to the student who wrote the best term paper in the first effort to use the [https://wright.miraheze.org/w/index.php?title=Main_Page&amp;oldid=672 Miraheze wikifarm] at [[Wright State University Lake Campus]] in the Spring of 2017.


==A thank-you note to ''Annie Isabelle Anonymous''==
[[File:Emoji u1f3c6.svg|left|75px|thumb|1&lt;sup&gt;st&lt;/sup&gt; prize]]
Dear A.I.A,

'''https://en.wikiversity.org/wiki/File:Anonymous_Life_in_the_Universe.pdf'''

Your Astronomy project resides at the link shown above.   I removed your real name and also password protected the pdf file.  It should be possible to edit it using an Adobe editor using the password ****.  You can use knowledge of that password to convince potential employers that you are the author of this impressive document.  

I will request that all future reports of this nature follow the format you invented:  A detailed summary of a chapter of a freely available textbook , with up  to 100 multiple choice questions embedded into the document (in the future, the answers will be in a key at the end of the document).  To be sure, many of your questions would be &quot;bad&quot; on an exam that would be used to assign a grade.  But, an instructor using a document like yours could point out the &quot;good&quot; questions during the lecture.  Or, the &quot;good&quot; questions could be extracted and stored on a private wiki to create a bank of questions that nobody else can see.  Or, the instructor could edit your text on a wiki to better suit his or her needs. Perhaps you (or somebody else) might wish to discuss this further at the talk page shown below.  

https://en.wikiversity.org/wiki/File_talk:Anonymous_Life_in_the_Universe.pdf

Also, in the future, it would be better if the document were on a private wiki where students can write a term paper without looking at each other's efforts, and create a document that would be easier for instructors to use and edit.  I am currently using Miraheze for this purpose. At the moment my little &quot;wikifarm&quot; looks like this:

https://wright.miraheze.org/w/index.php?title=Main_Page&amp;oldid=672

Let me know if you need a letter of recommendation.  And, &quot;Annie&quot;: THANK YOU !!!

{{subpages/List}}

Yours truly --[[User:Guy vandegrift]]

==Footnotes==
We need a repository for the open bank of quiz questions, and at the moment I see three options:
===[[Quizbank/Entire_bank]]===

'''Pros: ''' Compatibility with all public and private wikis, and the ease with which anybody can use Python to create and edit these exams. The &quot;conceptual&quot; questions use the wikimedia quiz extension, but the &quot;numerical&quot; questions would require a high-level language to create the wikitext.  These numerical questions typically come in 10 or more versions that have randomized numerical values.  I currently use Matlab to create these exams, but will soon switch over to Python (which I recently learned).

'''Cons: ''' '''''[[Quizbank]]''''' creates exams with a rather amateurish format.  They are rendered on a wiki and printed out as a pdf file.  For example, '''''[[Quizbank/Sample_rendition]]''''' can be be printed to create four midterms and a final exam (in two versions) for a conceptual course in physics.  Students can use this  [[Wright State University Lake Campus/2016-9/Phy1050/Study guide|'''''this  study guide''''']] to prepare for the exams.  A more serious first-year introductory physics course requires numerical questions where the students practice using different numerical values.  A study guide for such a course can be found [[Wright_State_University_Lake_Campus/2016-9/Phy2410/Study_guide|'''here''']].

==OpenStax True False progress==
By the end of summer I hope to have a large collection of simple reading questions for this book.  If you want to contribute leave a message at [[User talk:Guy vandegrift]].
{| class=&quot;wikitable&quot;
|  ||  ||  || 63 || 207 || Nom || Nom || 2.2 || 0.7
|-
|  ||  || pp || Sections || Questions || Sec || Ques || pp/sec || pp/ques
|-
| 1 Science and the Universe: A Brief Tour 11 || 11 || 2 ||  ||  || 0.928998009 || 2.849829391 ||  || 
|-
| 1.1 The Nature of Astronomy 13 || 13 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 1.2 The Nature of Science 13 || 13 || 2 ||  ||  || 0.928998009 || 2.849829391 ||  || 
|-
| 1.3 The Laws of Nature 15 || 15 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 1.4 Numbers in Astronomy 15 || 15 || 2 ||  ||  || 0.928998009 || 2.849829391 ||  || 
|-
| 1.5 Consequences of Light Travel Time 17 || 17 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 1.6 A Tour of the Universe 18 || 18 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 1.7 The Universe on the Large Scale 23 || 23 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 1.8 The Universe of the Very Small 27 || 27 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 1.9 A Conclusion and a Beginning 28 || 28 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 2 Observing the Sky: The Birth of Astronomy 31 || 31 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 2.1 The Sky Above 32 || 32 || 10 ||  ||  || 4.644990046 || 14.24914695 ||  || 
|-
| 2.2 Ancient Astronomy 42 || 42 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 2.3 Astrology and Astronomy 49 || 49 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 2.4 The Birth of Modern Astronomy 54 || 54 || 15 ||  ||  || 6.96748507 || 21.37372043 ||  || 
|-
| 3 Orbits and Gravity 69 || 69 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 3.1 The Laws of Planetary Motion 70 || 70 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 3.2 Newton’s Great Synthesis 76 || 76 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 3.3 Newton’s Universal Law of Gravitation 81 || 81 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 3.4 Orbits in the Solar System 85 || 85 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 3.5 Motions of Satellites and Spacecraft 88 || 88 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 3.6 Gravity with More Than Two Bodies 91 || 91 || 12 ||  ||  || 5.573988056 || 17.09897634 ||  || 
|-
| 4 Earth, Moon, and Sky 103 || 103 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 4.1 Earth and Sky 104 || 104 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 4.2 The Seasons 107 || 107 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 4.3 Keeping Time 114 || 114 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 4.4 The Calendar 117 || 117 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 4.5 Phases and Motions of the Moon 120 || 120 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 4.6 Ocean Tides and the Moon 125 || 125 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 4.7 Eclipses of the Sun and Moon 129 || 129 || 16 ||  ||  || 7.431984074 || 22.79863513 ||  || 
|-
| 5 Radiation and Spectra 145 || 145 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 5.1 The Behavior of Light 146 || 146 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 5.2 The Electromagnetic Spectrum 153 || 153 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 5.3 Spectroscopy in Astronomy 161 || 161 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 5.4 The Structure of the Atom 166 || 166 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 5.5 Formation of Spectral Lines 172 || 172 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 5.6 The Doppler Effect 176 || 176 || 13 ||  ||  || 6.03848706 || 18.52389104 ||  || 
|-
| 6 Astronomical Instruments 189 || 189 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 6.1 Telescopes 190 || 190 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 6.2 Telescopes Today 196 || 196 || 10 ||  ||  || 4.644990046 || 14.24914695 ||  || 
|-
| 6.3 Visible-Light Detectors and Instruments 206 || 206 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 6.4 Radio Telescopes 210 || 210 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 6.5 Observations outside Earth’s Atmosphere 217 || 217 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 6.6 The Future of Large Telescopes 222 || 222 || 11 ||  ||  || 5.109489051 || 15.67406165 ||  || 
|-
| 7 Other Worlds: An Introduction to the Solar System 233 || 233 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 7.1 Overview of Our Planetary System 234 || 234 || 12 || 3 || 25 || 5.573988056 || 17.09897634 || 4.0 || 0.5
|-
| 7.2 Composition and Structure of Planets 246 || 246 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 7.3 Dating Planetary Surfaces 251 || 251 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 7.4 Origin of the Solar System 254 || 254 || 11 ||  ||  || 5.109489051 || 15.67406165 ||  || 
|-
| 8 Earth as a Planet 265 || 265 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 8.1 The Global Perspective 266 || 266 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 8.2 Earth’s Crust 270 || 270 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 8.3 Earth’s Atmosphere 278 || 278 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 8.4 Life, Chemical Evolution, and Climate Change 283 || 283 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 8.5 Cosmic Influences on the Evolution of Earth 288 || 288 || 15 ||  ||  || 6.96748507 || 21.37372043 ||  || 
|-
| 9 Cratered Worlds 303 || 303 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 9.1 General Properties of the Moon 303 || 303 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 9.2 The Lunar Surface 310 || 310 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 9.3 Impact Craters 315 || 315 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 9.4 The Origin of the Moon 320 || 320 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 9.5 Mercury 321 || 321 || 14 ||  ||  || 6.502986065 || 19.94880573 ||  || 
|-
| 10 Earthlike Planets: Venus and Mars 335 || 335 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 10.1 The Nearest Planets: An Overview 335 || 335 || 7 || 3 || 15 || 3.251493033 || 9.974402867 || 2.3 || 0.5
|-
| 10.2 The Geology of Venus 342 || 342 || 5 || 6 || 28 || 2.322495023 || 7.124573477 || 0.8 || 0.2
|-
| 10.3 The Massive Atmosphere of Venus 347 || 347 || 3 || 3 || 19 || 1.393497014 || 4.274744086 || 1.0 || 0.2
|-
| 10.4 The Geology of Mars 350 || 350 || 9 ||  ||  || 4.180491042 || 12.82423226 ||  || 
|-
| 10.5 Water and Life on Mars 359 || 359 || 12 ||  ||  || 5.573988056 || 17.09897634 ||  || 
|-
| 10.6 Divergent Planetary Evolution 371 || 371 || 10 ||  ||  || 4.644990046 || 14.24914695 ||  || 
|-
| 11 The Giant Planets 381 || 381 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 11.1 Exploring the Outer Planets 381 || 381 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 11.2 The Giant Planets 387 || 387 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 11.3 Atmospheres of the Giant Planets 393 || 393 || 16 || 5 || 20 || 7.431984074 || 22.79863513 || 3.2 || 0.8
|-
| 12 Rings, Moons, and Pluto 409 || 409 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 12.1 Ring and Moon Systems Introduced 410 || 410 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 12.2 The Galilean Moons of Jupiter 411 || 411 || 9 || 4 || 8 || 4.180491042 || 12.82423226 || 2.3 || 1.1
|-
| 12.3 Titan and Triton 420 || 420 || 5 || 5 || 10 || 2.322495023 || 7.124573477 || 1.0 || 0.5
|-
| 12.4 Pluto and Charon 425 || 425 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 12.5 Planetary Rings 432 || 432 || 19 ||  ||  || 8.825481088 || 27.07337921 ||  || 
|-
| 13 Comets and Asteroids: Debris of the Solar System 451 || 451 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 13.1 Asteroids 452 || 452 || 11 || 7 || 16 || 5.109489051 || 15.67406165 || 1.6 || 0.7
|-
| 13.2 Asteroids and Planetary Defense 463 || 463 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 13.3 The “Long-Haired” Comets 466 || 466 || 10 || 5 || 10 || 4.644990046 || 14.24914695 || 2.0 || 1.0
|-
| 13.4 The Origin and Fate of Comets and Related Objects 476 || 476 || 17 ||  ||  || 7.896483079 || 24.22354982 ||  || 
|-
| 14 Cosmic Samples and the Origin of the Solar System 493 || 493 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 14.1 Meteors 494 || 494 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 14.2 Meteorites: Stones from Heaven 499 || 499 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 14.3 Formation of the Solar System 504 || 504 || 6 || 5 || 12 || 2.786994028 || 8.549488172 || 1.2 || 0.5
|-
| 14.4 Comparison with Other Planetary Systems 510 || 510 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 14.5 Planetary Evolution 515 || 515 || 12 ||  ||  || 5.573988056 || 17.09897634 ||  || 
|-
| 15 The Sun: A Garden-Variety Star 527 || 527 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 15.1 The Structure and Composition of the Sun 528 || 528 || 11 || 7 || 18 || 5.109489051 || 15.67406165 || 1.6 || 0.6
|-
| 15.2 The Solar Cycle 539 || 539 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 15.3 Solar Activity above the Photosphere 544 || 544 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 15.4 Space Weather 548 || 548 || 15 ||  ||  || 6.96748507 || 21.37372043 ||  || 
|-
| 16 The Sun: A Nuclear Powerhouse 563 || 563 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 16.1 Sources of Sunshine: Thermal and Gravitational Energy 563 || 563 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 16.2 Mass, Energy, and the Theory of Relativity 566 || 566 || 10 || 5 || 11 || 4.644990046 || 14.24914695 || 2.0 || 0.9
|-
| 16.3 The Solar Interior: Theory 576 || 576 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 16.4 The Solar Interior: Observations 582 || 582 || 13 ||  ||  || 6.03848706 || 18.52389104 ||  || 
|-
| 17 Analyzing Starlight 595 || 595 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 17.1 The Brightness of Stars 595 || 595 || 4 || 1 || 3 || 1.857996019 || 5.699658781 || 4.0 || 1.3
|-
| 17.2 Colors of Stars 599 || 599 || 3 || 1 || 3 || 1.393497014 || 4.274744086 || 3.0 || 1.0
|-
| 17.3 The Spectra of Stars (and Brown Dwarfs) 602 || 602 || 7 || 3 || 9 || 3.251493033 || 9.974402867 || 2.3 || 0.8
|-
| 17.4 Using Spectra to Measure Stellar Radius, Composition, and Motion 609 || 609 || 16 ||  ||  || 7.431984074 || 22.79863513 ||  || 
|-
| 18 The Stars: A Celestial Census 625 || 625 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 18.1 A Stellar Census 625 || 625 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 18.2 Measuring Stellar Masses 629 || 629 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 18.3 Diameters of Stars 636 || 636 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 18.4 The H–R Diagram 641 || 641 || 18 ||  ||  || 8.360982084 || 25.64846452 ||  || 
|-
| 19 Celestial Distances 659 || 659 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 19.1 Fundamental Units of Distance 659 || 659 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 19.2 Surveying the Stars 663 || 663 || 9 ||  ||  || 4.180491042 || 12.82423226 ||  || 
|-
| 19.3 Variable Stars: One Key to Cosmic Distances 672 || 672 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 19.4 The H–R Diagram and Cosmic Distances 679 || 679 || 12 ||  ||  || 5.573988056 || 17.09897634 ||  || 
|-
| 20 Between the Stars: Gas and Dust in Space 691 || 691 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 20.1 The Interstellar Medium 692 || 692 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 20.2 Interstellar Gas 696 || 696 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 20.3 Cosmic Dust 704 || 704 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 20.4 Cosmic Rays 711 || 711 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 20.5 The Life Cycle of Cosmic Material 714 || 714 || 2 ||  ||  || 0.928998009 || 2.849829391 ||  || 
|-
| 20.6 Interstellar Matter around the Sun 716 || 716 || 11 ||  ||  || 5.109489051 || 15.67406165 ||  || 
|-
| 21 The Birth of Stars and the Discovery of Planets outside the Solar System 727 || 727 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 21.1 Star Formation 728 || 728 || 9 ||  ||  || 4.180491042 || 12.82423226 ||  || 
|-
| 21.2 The H–R Diagram and the Study of Stellar Evolution 737 || 737 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 21.3 Evidence That Planets Form around Other Stars 740 || 740 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 21.4 Planets beyond the Solar System: Search and Discovery 744 || 744 || 9 ||  ||  || 4.180491042 || 12.82423226 ||  || 
|-
| 21.5 Exoplanets Everywhere: What We Are Learning 753 || 753 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 21.6 New Perspectives on Planet Formation 759 || 759 || 12 ||  ||  || 5.573988056 || 17.09897634 ||  || 
|-
| 22 Stars from Adolescence to Old Age 771 || 771 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 22.1 Evolution from the Main Sequence to Red Giants 772 || 772 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 22.2 Star Clusters 778 || 778 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 22.3 Checking Out the Theory 782 || 782 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 22.4 Further Evolution of Stars 789 || 789 || 9 ||  ||  || 4.180491042 || 12.82423226 ||  || 
|-
| 22.5 The Evolution of More Massive Stars 798 || 798 || 11 ||  ||  || 5.109489051 || 15.67406165 ||  || 
|-
| 23 The Death of Stars 809 || 809 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 23.1 The Death of Low-Mass Stars 810 || 810 || 5 ||  ||  || 2.322495023 || 7.124573477 ||  || 
|-
| 23.2 Evolution of Massive Stars: An Explosive Finish 815 || 815 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 23.3 Supernova Observations 822 || 822 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 23.4 Pulsars and the Discovery of Neutron Stars 830 || 830 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 23.5 The Evolution of Binary Star Systems 836 || 836 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 23.6 The Mystery of the Gamma-Ray Bursts 839 || 839 || 18 ||  ||  || 8.360982084 || 25.64846452 ||  || 
|-
| 24 Black Holes and Curved Spacetime 857 || 857 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 24.1 Introducing General Relativity 857 || 857 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 24.2 Spacetime and Gravity 863 || 863 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 24.3 Tests of General Relativity 866 || 866 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 24.4 Time in General Relativity 869 || 869 || 2 ||  ||  || 0.928998009 || 2.849829391 ||  || 
|-
| 24.5 Black Holes 871 || 871 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 24.6 Evidence for Black Holes 879 || 879 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 24.7 Gravitational Wave Astronomy 882 || 882 || 13 ||  ||  || 6.03848706 || 18.52389104 ||  || 
|-
| 25 The Milky Way Galaxy 895 || 895 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 25.1 The Architecture of the Galaxy 896 || 896 || 9 ||  ||  || 4.180491042 || 12.82423226 ||  || 
|-
| 25.2 Spiral Structure 905 || 905 || 4 ||  ||  || 1.857996019 || 5.699658781 ||  || 
|-
| 25.3 The Mass of the Galaxy 909 || 909 || 2 ||  ||  || 0.928998009 || 2.849829391 ||  || 
|-
| 25.4 The Center of the Galaxy 911 || 911 || 7 ||  ||  || 3.251493033 || 9.974402867 ||  || 
|-
| 25.5 Stellar Populations in the Galaxy 918 || 918 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 25.6 The Formation of the Galaxy 921 || 921 || 14 ||  ||  || 6.502986065 || 19.94880573 ||  || 
|-
| 26 Galaxies 935 || 935 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|-
| 26.1 The Discovery of Galaxies 936 || 936 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 26.2 Types of Galaxies 939 || 939 || 6 ||  ||  || 2.786994028 || 8.549488172 ||  || 
|-
| 26.3 Properties of Galaxies 945 || 945 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 26.4 The Extragalactic Distance Scale 948 || 948 || 3 ||  ||  || 1.393497014 || 4.274744086 ||  || 
|-
| 26.5 The Expanding Universe 951 || 951 || 14 ||  ||  || 6.502986065 || 19.94880573 ||  || 
|-
| 27 Active Galaxies, Quasars, and Supermassive Black Holes 965 || 965 || 0 ||  ||  || 0 || 0 ||  || 
|-
| 27.1 Quasars 965 || 965 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 27.2 Supermassive Black Holes: What Quasars Really Are 973 || 973 || 8 ||  ||  || 3.715992037 || 11.39931756 ||  || 
|-
| 27.3 Quasars as Probes of Evolution in the Universe 981 || 981 || 14 ||  ||  || 6.502986065 || 19.94880573 ||  || 
|-
| 28 The Evolution and Distribution of Galaxies 995 || 995 || 1 ||  ||  || 0.464499005 || 1.424914695 ||  || 
|}

===[https://www.osmosis.org/home/search Osmosis]===

'''Pros: ''' Much better developed, larger in scope, and more professional.

'''Cons: ''' At the moment only does medical board exams. Costs money for the student.

===[http://www.testpreppractice.net/CLEP/Physics-CLEP-Practice-Tests.html www.testpreppractice.net/CLEP/Physics-CLEP-Practice-Tests]===
'''Pros: ''' It's already been done! (but I just learned about it)

'''Cons: ''' We don't have a textbook (something for Wikiversity to fix?)

[[Category:Quizbank]]
[[Category:OpenStax]]</text>
      <sha1>4zgnfnkg66rgn423409m4zkcxsivcbf</sha1>
    </revision>
  </page>
  <page>
    <title>OpenStax University Physics</title>
    <ns>0</ns>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="3815" sha1="r9o3mziocvd3xwgpnhk88ynwhufj27e" xml:space="preserve">
[[File:Openstax University Physics Volume 1-LR.pdf|175px|right]]
[[File:University Physics Volume 2-LR 20161006.pdf|175px|right]]
[[File:UniversityPhysicsVolume3-LR.pdf|175px|right]]
&lt;small&gt;[[OpenStax]] | [[Quizbank]] | [[Physics equations]] | [[/Quizbank attribution/]]&lt;/small&gt;

This page is dedicated to the [[OpenStax]] textbook on the calculus-based Unviversity Physics.  The algebra/trig bases textbook is [[OpenStax College Physics]].


==Volume 2: Classroom-ready quizzes and exams are available on Wikiversity==
*'''[[Quizbank/Electricity and Magnetism (calculus based)]]: ''' Short quizzes based on chapter examples in Volume 1 Unit 2.
*'''[[Quizbank/Electricity and Magnetism (calculus) Exams]]: ''' Two 45 minute exams based on the aforementioned quizzes.
*'''[[Quizbank/University Physics Semester 2]]: ''' Supplements the previous two exam banks to include optics for a one-semester course.

==Equation sheets and summaries are available for much of this book==
'''Volume 1:'''
{{Special:PrefixIndex/OpenStax University Physics/V1}}
'''Volume 2:'''
{{Special:PrefixIndex/OpenStax University Physics/V2}}
'''Electromagnetism:'''
*[[Quizbank/Electricity and Magnetism (calculus based)/Equations]]

==Accessing the textbook==
*'''Download pdf files''' {{spaces|3}} [https://upload.wikimedia.org/wikiversity/en/4/4e/Openstax_University_Physics_Volume_1-LR.pdf Volume_1] {{spaces|3}} [https://upload.wikimedia.org/wikiversity/en/3/3e/University_Physics_Volume_2-LR_20161006.pdf Volume_2] {{spaces|3}} [https://upload.wikimedia.org/wikiversity/en/6/62/UniversityPhysicsVolume3-LR.pdf Volume_3]
*'''Link to same pdf files''' {{spaces|3}} [[:file:Openstax_University_Physics_Volume_1-LR.pdf|Volume_1]] {{spaces|3}} [[:file:University_Physics_Volume_2-LR_20161006.pdf|Volume_2]] {{spaces|3}} [[:file:UniversityPhysicsVolume3-LR.pdf|Volume_3]]

These are probably not the latest version.  The following online versions are probably newer:

*Volume 1:https://cnx.org/contents/1Q9uMg_a@11.1:Gofkr9Oy@17/Preface
*Volume 2: https://cnx.org/contents/eg-XcBxE@10.1:Gofkr9Oy@18/Preface
*Volume 3: https://cnx.org/contents/rydUIGBQ@10.1:Gofkr9Oy@19/Preface

While the latest version is typically the &quot;best&quot;, some (but certainly not all) instructors may prefer to keep an older version until all changes have been incorporated into the course. Wikiversarians should feel free to update these links.  Instructors who wish to &quot;freeze&quot; a certain edition need to place those links  on a different Wikiversity page with instructions to not upgrade.February 2018 (UTC)
&lt;noinclude&gt;

==Power point slides converted to PDF format for easy online use==
These slides were provided by [https://www.oercommons.org/groups/openstax-university-physics/1713/?__hub_id=27 OER].  '''The titles do not exactly match the chapters'''
===Volume 1===
[[c:File:MUPHYS1 00 Physics 1 (Course Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 01 Units and Measurements (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 02 Vectors (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 03 Motion Along a Straight Line (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 04 Motion in Two and Three Dimensions (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 05 Newton s Laws of Motion (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 06 Applications of Newton s Laws (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 07 Work and Kinetic Energy (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 08 Potential Energy and Convservation of Energy (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 09 Linear Momentum and Collisions (Module Overview).pdf]]&lt;br&gt;
[[c:File:MUPHYS1 10 Fixed Axis Rotation (Module Overview).pdf]]&lt;br&gt;
11-18 under construction.

====Volume 2====
[[c:File:MUPHYS2 00 Physics 2 (Course Overview).pdf]]

==Subpages==
{{Subpages/List}}
[[category:Quizbank]]
[[Category:OpenStax]]
&lt;/noinclude&gt;</text>
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  <page>
    <title>Open questions</title>
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      <timestamp>2026-09-03T04:17:13Z</timestamp>
      <contributor>
        <username>Michael Ten</username>
        <id>654933</id>
      </contributor>
      <comment>adde dsicuss qquwetions questions.. added essay ideas and ai prompt ideas.</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="701" sha1="sw92bckd82mx7dfo8f9dwdpkn5xp8en" xml:space="preserve">This is a place on this wiki for potentially unanswered questions that may have some educational value. These can be spun off into unique educational resources if appropriate. 
{{RightTOC}}

==After life==
Can science prove or disprove an after life? 

==Spirit==
Where does spirituality and science intersect? How? 

==Miscellaneous==
* What are some important unanswered questions within the field of cosmology?

== Discussion questions, essay ideas, and AI prompt ideas ==
* What sort of AI prompts might be good for an educational wiki? Be super meta. Give good prompts for an educational wiki. 

==Questions==
* [[Why do plants store food?]]
* [[Should suicide be legal?]]

[[Category:Questions]]</text>
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  <page>
    <title>Social Victorians/People/Gwladys Robinson</title>
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        <id>1331941</id>
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      <text bytes="109852" sha1="h9qu3kiue8oxo181a66v5upeipqyyhi" xml:space="preserve">{{Short description|Dress worn by Queen Victoria at her wedding to Prince Albert in 1840}}
= Sandbox =
Page to draft revisions for Wikipedia articles.

For Gwladys Robinson, see Gwladys Lowther Robinson, [[Social Victorians/People/Ripon|Marchioness of Ripon]] and, earlier, [[Social Victorians/People/Lowther|Countess of Lonsdale]]
==References==
{{reflist|2}}

[[Category:1840 works]]
[[Category:Royal wedding dresses|Victoria Queen]]
[[Category:1840s fashion]]
[[Category:British royal attire]]
[[Category:Dresses in the Royal Collection of the United Kingdom|Victoria, Wedding]]
[[Category:Diamond Jubilee of Queen Victoria]]

= Victorian fashion =

==Women's fashion==
== Hats and headwear ==
[[File:Ford.madox.brown.last.emma.study.jpg|thumb|''Emma Hill'' by [[Ford Madox Brown]] (1853), a woman wearing a later version of the [[poke bonnet]]]]
[[File:Hoed,_objectnr_KA_1237.tif|left|thumb|Perched bonnet style of the early 1870s.]]
Hats were crucial to a respectable appearance for both men and women. 

=== Men's Hats ===
The top hat, for example, was standard formal wear for upper- and middle-class men.[Payne] According to Blanche Payne, &quot;The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century.&quot; (457–58) 

Although top hats were the dominant hat in the 19th century, other hats became popular for working classes and lower income middle class. “The style of an individual’s hat varied, depending on fashion and their social position, as well as their profession or chosen activity..” ''Goodman 53 of 460'' Other hats that became necessary and popular include the Derby, the straw Boater, and a flat cap with a short brim. In many cases the class, work activity and income could be determined by what kind of hat was on the head of the wearer. For some men, a hat supporting a particular sport or team was important.

The Derby or Bowler hat was designed by William and Thomas Bowler, brother shopkeepers in 1849. ''(Goodman 55 of 460)'' It cost less than a top hat but  lasted longer and was soon worn by middle class bankers and clerks. Straw boaters were worn by the aristocracy for casual events and working class factory workers and agricultural laborers. By 1901 working class men had changed their preference to the flat caps which became the most popular hat for the workers. 

Headdress for men was an essential part of dress for the entire period of the Victorian age, from the 1830s through the end of the century. Judith Flanders describes the hats worn by men in London, &lt;blockquote&gt;It is difficult to bear in mind the importance of hats as not only markers of class and income, but also as indicators of respectability. [509–519] [George Augustus] Sala commented that &quot;every&quot; man throughout the history of the world &quot;must, necessarily and habitually, wear some kind of covering to his head&quot;. Postmen wore hats, small children wore hats, field labourers and market gardeners wore hats, cricketers, skaters — all sportsmen — wore hats. It was, self-evidently, impossible to go outdoors without one. ... Those in professional occupations wore pot hats, as did clerks and all those with pretensions to middle-class status. Even doctors' delivery boys wore battered hand-me-down pot hats: &quot;the nap rusty, the band a mournful strip of tarnished lace; but still a Hat&quot;, which &quot;stamps him as being associated, in however slender a manner, with a learned profession&quot;. Cloth caps were for labourers, for costers and for boys. ... Artisans wore caps made out of paper, which they folded [510–511] themselves and so could easily replace as they became dirty.&lt;ref name=&quot;:23&quot;&gt;{{Cite book|title=The Victorian City: Everyday Life in Dickens' London|last=Flanders|first=Judith|publisher=Thomas Dunne Books|year=2012|location=New York, New York}}&lt;/ref&gt; (509–511 [of 972]) &lt;/blockquote&gt;A pot-hat is a kind of derby or bowler in men's hats. (Lewandowski, 237)

==== Original Text ====
Hats were crucial to a respectable appearance for both men and women. The top hat, for example, was standard formal wear for upper- and middle-class men.&lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Steele |first=Valerie |url=https://archive.org/details/fashioneroticism0000stee |title=Victorian Fashion. Fashion and Eroticism: Ideals of Feminine Beauty from the Victorian Era to the Jazz Age |publisher=Oxford University Press |year=1985 |isbn=978-0-19-503530-8 |pages=[https://archive.org/details/fashioneroticism0000stee/page/51 51]–84 |url-access=registration}}&lt;/ref&gt; For women, the styles of hats changed over time and were designed to match their outfits.

=== Women's Hats ===
For a discussion of the history of plumes and feathers, see [[Social Victorians/Victorian Things#Ostrich Plumes and Prince of Wales's Feathers|Ostrich Plumes and Prince of Wales's Feathers in ''Victorian Things'']].

==== Original Wikipedia Text ====
During the early Victorian decades, hats were modest in size and design, straw and fabric bonnets being the popular choice. [[Poke bonnet]]s, which had been worn during the late [[Regency period]], had high, small crowns and brims that grew larger until the 1830s, when the face of a woman wearing a poke bonnet could only be seen directly from the front. They had rounded brims, echoing the rounded form of the bell-shaped hoop skirts.

Bonnets shrunk at the end of the 1860s and moved to a perched position in the early 1870s as hairstyles grew in scale and intricacy. This led to the popularization of hats, which became the headwear of choice for the remainder of the Victorian era.&lt;ref name=&quot;g4223&quot;&gt;{{cite book |last=Cunnington |first=Cecil Willett |title=English Women's Clothing in the Nineteenth Century |date=1990-05-01 |publisher=Courier Corporation |isbn=0-486-26323-1 |publication-place=New York |page=}}&lt;/ref&gt;
[[File:The_London_and_Paris_ladies'_magazine_(Apr_1885)_03.png|thumb|Flower pot style hat of 1885.]]
The 1880s saw a hat inspired by the top hat for women known as the flowerpot hat, and the 1890s saw the popularity of the boater. The hats of the late Victorian era were covered with elaborate creations of silk flowers, ribbons, and above all, exotic plumes; hats sometimes included entire exotic birds that had been stuffed. Many of these plumes came from birds in the Florida everglades, which were nearly made entirely extinct by overhunting. By 1899, early environmentalists like [[Adeline Knapp]] were engaged in efforts to curtail the hunting for plumes. By 1900, more than five million birds a year were being slaughtered, and nearly 95 per cent of Florida's shore birds had been killed by [[Plume hunting|plume hunter]]s.&lt;ref&gt;{{cite web|title=Everglades National Park|url=https://www.pbs.org/nationalparks/parks/everglades/|archive-url=https://web.archive.org/web/20090927085907/http://www.pbs.org/nationalparks/parks/everglades/|url-status=dead|archive-date=27 September 2009|publisher=PBS|access-date=7 November 2011}}&lt;/ref&gt;

== Shoes ==
The women's shoes of the early Victorian period were narrow and heelless, in black or white satin. By 1850s and 1860s, they were slightly broader with a low heel and made of leather or cloth. Ankle-length laced or buttoned boots were also popular. From the 1870s to the twentieth century, heels grew higher and toes more pointed. Low-cut pumps were worn for the evening.&lt;ref name=&quot;:4&quot; /&gt;

== Cosmetics ==
[[Victorian-era cosmetics]] were typically less obvious than ours. However, small amounts of pale face powder or powdered blush were widely used.&lt;ref&gt;{{Cite book |last=Goodman |first=Ruth |title=How to be a Victorian |date=2014 |publisher=Penguin Books |isbn=978-0-670-92136-2 |location=London}}&lt;/ref&gt; Tints were sometimes added to face creams. Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}.

Hair color

== Men's fashion ==
(Some of this belongs in the general intro to both women's and men's fashion, and some should go here?)
* evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing. For the growing and rising middle classes, gender roles became more and more rigid around the concept of the separate spheres, the public for men and the private sphere for women. 
* Shirts and collars separated: &quot;by 1827 detachable collars became available&quot;&lt;ref name=&quot;:25&quot;&gt;{{Cite book|title=The History of Costume: From Ancient Mesopotamia Through the Twentieth Century|last=Payne|first=Blanche|last2=Winakor|first2=Geitel|last3=Farrell-Beck|first3=Jane|publisher=Addison-Wesley Longman|year=1992|isbn=0-06-047141-7|edition=2nd|location=New York, New York}}&lt;/ref&gt; (477)
* Beau Brummell: &quot;Men began to aspire to fine cutting, tailoring, and perfect fit in their clothes, flawless grooming and manners in themselves. The Englishman George Bryan (Beau) Brummell deserves much credit for the ideal of meticulous masculine appearance.&quot;&lt;ref name=&quot;:25&quot; /&gt; (458) &quot;From 1796 to 1816, the Beau set the pattern for cleanliness and liberal use of starch.&quot; (459)
* Early 19th-c: &quot;Tailored wool garments, originally inspired by the clothing of English country gentry, formed the backbone of men's wardrobes in the early nineteenth century.&quot;&lt;ref name=&quot;:25&quot; /&gt; (458)
* The most sweeping change in men’s fashion before 1820 was the length of their pants. Trousers that went from waist to foot were so much more comfortable and popular than knee breeches that they replaced what had dominated men’s wear since the sixteenth century. And trousers have been worn for over two centuries now.
* &quot;During the 1820s and 1830s, men sometimes wore two waistcoats, in combinations like white velvet over rose-and-gold brocade.&quot;&lt;ref name=&quot;:25&quot; /&gt; (474) Men's waistcoats (or vests) could be colorful and embellished so that they were the focus of the ensemble of dark colored fabric of the jacket and trousers. By the end of the 19th century waistcoats made from the same fabric as the trousers and coat dominated men’s fashion.
Cultural sources for men's clothing and men's fashion: industrial revolution; expanding middle class; technological advancements; an association in the culture between outward appearance and inward nature; French Revolution; Albert Edward, Prince of Wales
Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):&lt;blockquote&gt;A study: (Historical, and Cultural Impact on the Costume Development) showed that depending on the functional and aesthetic characteristics, the division of clothing according to gender and age continued for centuries, whether informal or ceremonial, and varied according to gender, general style, nature of the jewelry, as well as family status, and stated that the traditional costume indirectly linked man to nature, as it was a gateway to the relationship between the body (the small world) and the world (the big world) [20].&lt;ref&gt;{{Cite journal|last=Alzahrani|first=Sarah Gharmallah|last2=Saroukh|first2=Safia Abdelaziz|date=2024|title=The Semiotic Dimension of Men's Fashion in Modern Eras|url=http://www.sciencepg.com/journal/ijla|journal=International Journal of Literature and Arts|volume=Vol. 12, No. 5|via=Research Gate}}&lt;/ref&gt; (136)

[20] Park, S.J., &amp; Park, K.S. (2006). Semiotic Analysis on Advertisement Expression of Men's Toiletries. The Research Journal of the Costume Culture, 14(2), 234-246.&lt;/blockquote&gt;

=== Industrial Revolution and Technological Advancements ===

* the railroad
* mass production of fabrics for working-class men's clothing
** Jacquard looms
**
* industrial sewing machines
* aniline dyes, including black

=== Expanding Middle Class ===

* gender roles getting more clearly defined and rigid (referring back to page overview)
* The growing middle class involved among other things more and more jobs and careers for young men as clerks, office workers, they were junior, subordinate, and they were commuting on the railroads from suburbs.

Carolyn Kirby:&lt;blockquote&gt;In western Europe the fashion for plain dark suits coincided with the rise of the affluent middle-classes in a world where the pace of industrialisation and the globalisation of trade was accelerating as never before. The sharp, dark business suit became the last word in male power-dressing. And so it remains to this day.&lt;ref&gt;{{Cite web|url=https://historiamag.com/invent-masculine-fashion/|title=The invention of masculine fashion|last=Kirby|first=Carolyn|date=3 December 2025|website=Historia: Magazine of the Historical Writers' Association|access-date=25 August 2026}}&lt;/ref&gt;&lt;/blockquote&gt;Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):&lt;blockquote&gt;The Industrial Revolution that began in the late 18th century had a great impact on the development of fashion in the 19th century, there was a clear change in men's clothing at the beginning of the 19th century, not only in style but also in the appearance of the English sewing machine, and from this date, English clothing became world-class, and this was not only for England but for all of Europe is undoubtedly due to the French Revolution that stripped Europe of its previous leadership of fashion, so the 19th century belonged to the English in terms of fashion [16].

[16]  Hussein, T. (2002). The History and Development of Fashion „Part III‟ Modern Times, Nahdet Misr for Printing and Publishing, Cairo.&lt;/blockquote&gt;

=== French Revolution ===
David Kuchta:&lt;blockquote&gt;... since 1666, male gentility has been associated with modesty and plainness in dress. Eschewing fashion as an increasingly feminized realm Charles II's vest inaugurated a new and essentially modern era of masculine aesthetics, one that reversed a long-held association between elaborate display and high social status. Manly thrift now displayed elite status.&lt;ref&gt;{{Cite book|title=The Three-Piece Suit and Modern Masculinity, England 1550–1850|last=Kutcha|first=David|publisher=University of California Press|year=2002|location=Berkeley and Los Angeles}}&lt;/ref&gt; (2)&lt;/blockquote&gt;Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):&lt;blockquote&gt;The 19th century started with a fashion landscape that was changing dramatically and rapidly from the styles of a generation earlier. The French Revolution brought fashions that had been emerging since the 1780s to the forefront. Neoclassicism now defined fashion as both men and women taking inspiration from classical antiquity. For women, the high-waisted silhouette in lightweight muslin was the dominant style, while fashionable men looked to the tailors of Britain for a new, refined look [17].

[17]  Franklin, H. (Aug 18, 2020). Published on Jun 25, 2020, Retrieved: &lt;nowiki&gt;https://fashionhistory.fitnyc.edu/1800-1809/&lt;/nowiki&gt; 11/11/2023. Edited. ...&lt;/blockquote&gt;Brent Shannon:&lt;blockquote&gt;&quot;Costume,&quot; wrote Max Beerbohm in 1896, &quot;enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not&quot; (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. &quot;Perhaps there is a tendency among Englishmen to judge a man too much by the shape of his hat or the kind of collar he wears,&quot; conduct author John Wanamaker confessed; &quot;But one must remember that in England if you ''wear'' the wrong thing, you will probably ''do'' the wrong thing, and generally ''be'' the wrong thing&quot; (1).&lt;sup&gt;11&lt;/sup&gt;

Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.&lt;ref&gt;{{Cite book|title=The Cut of His Coat: Men, Dress, and Consumer Culture in Britain, 1860–1914|last=Shannon|first=Brent Alan|publisher=Ohio University Press|year=2006|location=Athens, Ohio}}&lt;/ref&gt; (148)&lt;/blockquote&gt;

=== Influence of Albert Edward, Prince of Wales ===
Albert Edward, Prince of Wales was very concerned with fashion and authoritative about it, with a very specific eye to small details.
McNeil:&lt;blockquote&gt;When that great lover of pleasure, Edward VII, visited Marienbad incognito as the Duke of Lancaster, he was followed by tailors from Paris, Budapest, Vienna, and Berlin who photographed him and took notes about his clothes. Edward VII introduced many [423–424] novelties into men’s fashion. For the countryside such as at Sandringham, he permitted an informal dress code. The Henry Poole ledger marked as “HRH 1865” is for an evening coat without tails, the first “dinner jacket.” He is also credited with making fashionable the creased trouser in 1909 (his groom dried them with a board weight after heavy rain, resulting in the line), turned-up cuff trouser (after hitching his trouser bot- toms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat.&lt;ref&gt;McNeil, Peter. &quot;Men's Fashion: 1800–2022.&quot; Chapter 22. ''The Routledge History of Fashion and Dress, 1800 to the Present''. Routledge, 2024. https://opus.lib.uts.edu.au/bitstream/10453/182707/2/Men%27s%20Fashion%20200822_24_12_20_09_29_44.pdf

DOI: 10.4324/9780429295607-27.&lt;/ref&gt;&lt;/blockquote&gt;Virginia Cowles:&lt;blockquote&gt;It would be wrong to give the impression that the Heir Apparent was unhappy. If he could not work, at least he could play, and he did this very well. He loved being royal. He revelled in the rank and authority and privilege and luxury that accompanied the role of Prince of Wales. There were radicals who liked to lampoon him, and courtiers who wanted to reform him. But there was a much bigger group, a rich, fashionable, powerful society who adored him, fawned on him, gratified him, and copied everything he did.

Paradoxically this adulation often increased the Prince’s freedom of movement. A contemporary writer states that it was possible for the Prince of Wales to walk along Piccadilly, or St. James’ Street or Pall Mall without being recognized. Why? Because photography was still undeveloped? Oh no. It was due to ‘the curious fact that there are in society several gentlemen who bear an extraordinary resemblance to him, and who take some pride in dressing and moving exactly like him, so that it is often very difficult to identify him as he passes in the street on foot or in a hansom cab.

But the vogue of imitating the Prince did not stop at his beard, his clothes and his walk. Once when he had an attack of rheumatism in his shoulder, he was obliged to shake hands with his '''expo''' pressed stiffly to his side. Immediately this peculiar hand-shake was adopted by fashionable London. And when Alexandra [128–129] had a severe illness in the late sixties which left her lame for life, the smartest ladies in the land began to walk with a slightly halting gait, which became known as ‘the Alexandra Limp’.

The aping of royalty was not considered vulgar. On the whole the Prince and Princess were amused and flattered by it, but every now and then someone went too far. On one occasion a rich manufacturer from the North drove in the Park with his horses wearing headbands of the royal scarlet used exclusively by the Prince. The Heir Apparent did not attempt to hide his displeasure. His blue eyes grew cold, and his lower lip protruded in the famous Guelph pout. As a sharp lesson to the perpetrators of this unforgivably bad taste he drove in the Park the next day with his horses wearing black headbands. The manufacturer’s wife and daughters could not fail to observe the significance of this slight, and left the Park in tears; and the Prince’s friends congratulated him on his clever rebuff.

The Prince was not just ‘a swell’. In the jargon of the day he was ‘a heavy swell’, and apparently there was a world of difference between the two terms. A swell was a rich young aristocrat who lived in extreme comfort; but a heavy swell added showmanship to the comfort and lived in a stylish luxury that even the French were obliged to envy. And of course the heavy swell was the acme of sartorial elegance.

The Prince did not mind changing his dress half a dozen times a day. He loved clothes, and since whatever he chose to wear became the prevailing fashion overnight, he soon was regarded as an expert on the subject. His tailor-in-chief made a fortune. For many years he patronised a Mr. Poole. He discovered this gentleman by accident. He went to the theatre one night to see a well-known actor by the name of Fecher playing ‘Robert Macaire’. As an impecunious adventurer [129–130] Fechter was obliged to wear a coat that was torn and dirty, but Bertie’s expert eye noticed the elegant cut. At the end of the performance he asked Fechter for the name of his tailor, and Mr. Poole’s future was assured.

The Prince had so many clothes he could never travel with less than two valets; and two more valets were left at home cleaning, brushing and pressing his vast wardrobe. There were suits and coats for every variation of every climate the world over. There were over a hundred pieces of headgear; and since Bertie was an honorary admiral and an honorary general of most of the countries of Europe, there was an entire room devoted to uniforms, sashes, epaulettes, belts, buckles, swords, feathers and other regalia.

As the years rolled on the Prince became an ever-increasing authority on dress. Tailors from all over Europe used to gather to study his clothes. Their favorite meeting place was Homburg, and later, Marienbad. Here they could catch a glimpse of the Prince half a dozen times a day, strolling along the promenade, or riding in an open carriage. Once Bertie dressed hurriedly and forgot to fasten the last button on his waistcoat; this became a permanent fashion.

British manufacturers were not slow to realise what an asset they had in the Heir Apparent and kept a vigilant eye on his movements. Once, one of them declared in outraged tones that he was buying his gloves in France. A storm blew up of such proportions that the Prince’s secretary, Sir Francis Knollys, was forced to make a statement to the press. First, he declared that the Prince always had his gloves made in England, and second (and this was calculated to silence the critics) that His Royal Highness was very economical in the use of gloves and only found it necessary to order two dozen pairs a year.

Men’s clothes became of such importance that new [130–131] shops sprang up like mushrooms in Savile Row, Clifford Street and Bond Street. Most of the Prince’s innovations were inspired by comfort and convenience. He altered the cut of the evening dress waistcoat, he shortened the tails on the tail coat, he left his frock coat open (due to an increasing girth), he introduced the black homburg, and he attended race meetings, not in the frock coat hitherto ''de regueur'' but in tweeds. He tried having his trousers creased down the sides rather than the front and back, in order to hide his bandy legs, but this idea did not catch on, and he soon discarded it himself. But the prince was not the only arbiter of men’s fashions. The band of  &quot;heavy swells&quot; who followed his lead gave him plenty of competition. Lord Raglan and Lord Petersham invented coats which are still named after them. Lord Dupplin the dinner jacket and Lord Cardigan the button-up sweater. But Lord Hardwicke made the most spectacular contribution. Men’s silk hats were made of beaver which was left in its original rough, shaggy state. Lord Hardwicke polished his hat until he could see his face in it, and consequently was known as &quot;Glossy Top&quot;. He is responsible for the top hat as we know it today.&lt;ref&gt;{{Cite book|title=Gay monarch, the life and pleasures of Edward VII|last=Cowles|first=Virginia|publisher=Harper|year=1956|location=New York, New York|archive-url=https://archive.org/details/gaymonarchlifepl0000cowl/}}&lt;/ref&gt; (128–131)&lt;/blockquote&gt;

=== Overview of What Victorian Men Wore ===
Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):&lt;blockquote&gt;Men's clothing during this century consisted of black, brown, blue (dark, shiny, or bright), olive green, and grey. The preferred beautiful colors for evening wear were blue, followed by brown and green, while the fabrics for summer trousers were dark grey or black (with blue coats), and for daywear were light colors such as white or beige (Hussein, T. 2002). [16].

[16]  Hussein, T. (2002). The History and Development of Fashion „Part III‟ Modern Times, Nahdet Misr for Printing and Publishing, Cairo.&lt;/blockquote&gt;

=== 1840s ===

==== Original Text ====
[[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]]

During the [[1840s in fashion|1840s]], men wore tight-fitting, calf length [[frock coat]]s and a [[waistcoat]] or vest. Sleeves were full at the top and waists were tight, creating an hourglass form. Waistcoats were single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. For more formal occasions, a cutaway morning coat was worn with light trousers during the daytime, and a dark tail coat and trousers was worn in the evening. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[Cravat (early)|cravat]]s or neck ties. Trousers had fly fronts, and [[breeches]] were used for formal functions and when horseback riding. Men wore [[top hat]]s, with wide brims in sunny weather.

=== 1850s ===

* transition from frock coats to ditto suits, 1850s (Payne, 463)&lt;br /&gt;
According to Judith Flanders,&lt;blockquote&gt;While hackney drivers were also considered to be stereotypically shabby, hansom-cab drivers were generally represented as smartly dressed. A print in 1850 showed a driver in a snappy brown coat instead of the coachman’s heavy multiple-caped outfit, pale green striped trousers, short boots and top hat, the [167–168] reins held daintily in his gloved hands. Both cab and coach drivers wore top hats, but cabbies of a sporting bent later switched to bowlers, and in summer donned bright checked outfits.&lt;ref name=&quot;:23&quot; /&gt; (167–168 [of 972])&lt;/blockquote&gt;

==== Original Text ====
During the [[1850s in fashion|1850s]], men started wearing shirts with high upstanding or turnover [[collar (clothing)|collars]] and [[necktie#Four-in-hand|four-in-hand necktie]]s tied in a bow, or tied in a knot with the pointed ends sticking out like &quot;wings&quot;. The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class.

=== 1860s ===

==== Original Text ====
In the [[1860s in fashion|1860s]], men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. Frock coats were shortened to knee-length and were worn for business, while the mid-thigh length [[sack coat]] slowly displaced the frock coat for less-formal occasions, with the overall effect of a looser silhouette. Top hats briefly became the very tall &quot;stovepipe&quot; shape, but a variety of other hat shapes were popular.

=== 1870s ===

* To correct and prevent errors being made in court dress, in 1875 the Lord Chamberlain published ''Dress Worn by Gentleman at Her Majesty's Court'', &quot;a summary of regulations for court uniform and dress.&quot;&lt;ref&gt;{{Cite book|url=https://www.google.com/books/edition/Dress_worn_by_Gentlemen_at_Her_Majesty_s/pvrbQCXq0MEC?hl=en|title=Dress worn by Gentlemen at Her Majesty's Court|last=Britain)|first=Victoria (Queen of Great|date=1875|language=en}}&lt;/ref&gt; This is the kind of thing Bertie really cared about.

==== Original Text ====
During the [[1870s in fashion|1870s]], three-piece suits grew in popularity along with patterned fabrics for shirts. Neckties were the four-in-hand and, later, the [[Ascot tie]]s. A narrow ribbon tie was an alternative for tropical climates, especially in the Americas. Both frock coats and sack coats became shorter and more form fitting. Flat straw boaters were worn when boating.

=== 1880s ===

==== Original Text ====
During the [[1880s in fashion|1880s]], formal evening dress remained a dark tail coat and trousers with a dark waistcoat, a white bow tie, and a shirt with a winged collar. In mid-decade, the dinner jacket or [[tuxedo]], was used in more relaxed formal occasions. The [[Norfolk jacket]] and tweed or woolen breeches were used for rugged outdoor pursuits such as shooting. Knee-length topcoats, often with contrasting velvet or fur collars, and calf-length overcoats were worn in winter. Men's shoes had higher heels and a narrow toe.

=== 1890s ===

==== Original Text ====
Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.&lt;ref&gt;{{cite web|last=Landow|first=George|url=http://www.victorianweb.org/art/costume/90s/2.html|title=Men's informal sporting dress, late 1880s and '90s}}&lt;/ref&gt;

Throughout much of the Victorian era most men wore fairly short hair.  This was often accompanied by various forms of facial hair including moustaches, side-burns, and full beards.  A clean-shaven face did not come back into fashion until the end of the 1880s and early 1890s.&lt;ref&gt;{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}&lt;/ref&gt;

Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.&lt;ref name=&quot;shannon597&quot;&gt;{{cite journal|last=Shannon|first=Brent|title=Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914|journal=Victorian Studies|year=2004|volume=46|issue=4|pages=597–630|doi=10.1353/vic.2005.0022}}&lt;/ref&gt;

=== Notes ===

*Men's suits buttoned higher up than today (Payne, 467)
* Norfolk jackets and sack suits (Payne, 471)
* formal attire, tuxedos with tails, cutaways (Payne, 469)
* Keith Middlemas (https://archive.org/details/storyoffiesta00huxf/page/200/mode/2up?q=fashion)
*Brent Shannon. &quot;Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914.&quot; Victorian Studies 46, no. 4 (Summer 2004): 597–630.
==Mourning black==
{{See also |Mourning stationery}}
[[File:The royal children in mourning Mar 1862.jpg|thumb|Victoria's five daughters (Alice, Helena, Beatrice, Victoria and Louise), photographed wearing mourning black beneath a bust of their late father, Prince Albert (1862)]]
[[File:Mourning dress MET 50.40.3a-b front CP4.jpg|alt=Black Victorian mourning dress|thumb|Mourning Dress, 1894–95]]

In Britain, black is the colour traditionally associated with mourning for the dead. The customs and etiquette expected of men, and especially women, were rigid but evolving during much of the Victorian era. The expectations depended on a complex hierarchy of close or distant relationship with the deceased. (Davidoff) The closer the relationship, the longer the mourning period and the wearing of black. The wearing of full black was known as First Mourning, which had its own expected attire, including fabrics, and an expected duration of 4 to 18 months. Following the initial period of First Mourning, the mourner would progress to Second Mourning, a transition period of wearing less black, which was followed by Ordinary Mourning, and then Half-mourning. Some of these stages of mourning were shortened or skipped completely if the mourner's relationship to the deceased was more distant. Half-mourning was a transition period when black was replaced by acceptable colours such as lavender and mauve, possibly considered acceptable transition colours because of the tradition of [[Church of England]] (and [[Catholic Church|Catholic]]) clergy wearing lavender or mauve [[Stole (vestment)|stoles]] for funeral services, to represent the [[Passion (Christianity)|Passion of Christ]].&lt;ref&gt;{{cite web|title=The Colors of the Church Year|url=http://fullhomelydivinity.org/articles/colors.htm|publisher=Consortium of Country Churches|access-date=6 November 2011|archive-date=13 November 2011|archive-url=https://web.archive.org/web/20111113075214/http://fullhomelydivinity.org/articles/colors.htm|url-status=dead}}&lt;/ref&gt;

The mourning dress worn by Queen Victoria (below, right) &quot;shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death.&quot;&lt;ref&gt;{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&amp;searchField=All&amp;sortBy=Relevance&amp;deptids=8&amp;ft=queen+victoria&amp;offset=0&amp;rpp=20&amp;amp;pos=2|title=Mourning Dress, 1894–95|last=The Metropolitan Museum of Art|date=7 September 2019|website=The Metropolitan Museum of Art|access-date=7 September 2019}}&lt;/ref&gt; Dating from 1894–95, Queen Victoria wore this dress as a result of the death of the eldest son of the Prince and Princess of Wales, Eddy, in line to the throne.

=== Norms for mourning===

''Manners and Rules of Good Society, or, Solecisms to be Avoided'' (London, Frederick Warne &amp; Co., 1887) gives clear instructions, such as the following:&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–83}}&lt;/ref&gt;

{| class=&quot;wikitable&quot;
|-
! Relationship to deceased !! First mourning !! Second mourning !! Ordinary mourning !! Half-mourning
|-
| Wife for husband || 1-year, 1-month; [[bombazine]] fabric covered with [[Crape|crepe]]; [[widow's cap]], [[lawn cuff]]s, collars || 6 months: less crepe || 6 months: no crepe, silk or wool replaces bombazine; in last 3 months jet jewellery and ribbons can be added || 6 months: colours permitted are grey, lavender, mauve, and black-and-grey
|-
| Daughter for parent || 6 months: black with black or white crepe (for young girls); no linen cuffs and collars; no jewellery for first 2 months || 4 months: less crepe ||  –  || 2 months as above
|-
| Wife for husband's parents || 18 months in black bombazine with crepe || – || 3 months in black || 3 months as above
|-
| Parent for son- or daughter-in-law's parent || – Black armband in representation of someone lost || – || 1-month black || –
|-
| Second wife for parent of a first wife || – || – || 3 months black || –
|}

The complexity of these etiquette rules extends to specific mourning periods and attire for siblings, step-parents, aunts and uncles distinguished by blood and by marriage, nieces, nephews, first and second cousins, children, infants, and &quot;connections&quot; (who were entitled to ordinary mourning for a period of &quot;1–3 weeks, depending on level of intimacy&quot;). Men were expected to wear mourning black to a lesser extent than women, and for a shorter mourning period. After the mid-19th century, men would wear a black hatband and black suit, but for only half the prescribed period of mourning expected of women. Widowers were expected to mourn for a mere three months, whereas the proper mourning period expected for widows was up to four years.&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–9}}&lt;/ref&gt; Women who mourned in black for longer periods were accorded great respect in public for their devotion to the departed, the most prominent example being Queen Victoria herself. It was not uncommon for a widow who did not remarry to wear half-mourning for the rest of her life, except when another death necessitated full mourning. For example, Alexandra, Princess of Wales wore half-mourning for the rest of her life after her eldest son Eddy died in 1894. Empress Elisabeth of Austria did the same, as did Empress Eugénie of France. They reverted to full mourning when appropriate, but they never wore less than half-mourning after their sons' deaths.

Women with lesser financial means tried to keep up with the example being set by the middle and upper classes by dyeing their daily dress. Dyers made most of their income during the Victorian period by dyeing clothes black for mourning.&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}&lt;/ref&gt;

== Technological advancement ==
The technological changes that affected the manufacture and consumption of clothing in the Victorian age included the following:

* the mass production of fabrics — for example, &quot;by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year&quot;  [62]
* the invention of aniline dyes, which were much more vibrantly colored and resistant to fading than the natural dyes that had been used. — . Invented by chemist [[William Henry Perkin]] in 1856, the first aniline dye mauveine (or mauve) &quot;wash[ed] the fashionable landscape in a haze of purple.&quot;&lt;ref name=&quot;:22&quot;&gt;{{Cite book|title=The Dress Diary: Secrets from a Victorian Woman's Wardrobe|last=Strasdin|first=Kate|publisher=Pegasus Books|year=2023|location=New York, New York}}&lt;/ref&gt; (247) Other intense and, to the Victorians, intensely exciting colors followed, but the new synthetic additions to fabric sometimes included chemicals harmful to their wearers. For example, a &quot;bright-magenta hue was achieved by adding arsenical-based chemicals to existing aniline dyes, brightening the already luminous shades – but these left residues themselves, along with a toxic labour trail in their wake.&quot;&lt;ref name=&quot;:22&quot; /&gt; (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: &quot;The craze for artificial foliage to adorn the heads and dresses of women of fashion in the mid-nineteenth century had seen the proliferation of flower workshops, where young women in their hundreds laboured to produce the lifelike green leaves and blooms that would make a fetching headdress or would trail becomingly across the bodice of a gown. The lushness of the green was achieved by the application of a powder, a pigment that was created by mixing copper and the highly toxic chemical, arsenic trioxide. The physical effects of working with this poisonous compound were horrific. Contemporary medical drawings depict the green hue of the skin and dreadful open lesions on the hands of the maker, whilst the daily gradual ingestion of the powder by the flower girls was eventually fatal.&quot;&lt;ref name=&quot;:22&quot; /&gt; (255)
* the invention of a sewing machine that could be used in the home. Although sewing machines were already in use in the clothing industry, in 1858 Isaac Merritt Singer began to sell &quot;lightweight domestic machines&quot; for home sewing, radically increasing women's control over their own dress.&lt;ref name=&quot;:24&quot;&gt;{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen &amp; Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}&lt;/ref&gt; (91 [of 298])
* the spread of journalism for women and fashion journalism
* the Jacquard loom: &quot;French inventor Joseph-Marie Jacquard’s loom attachment mechanized the weaving of figured fabrics by 1804.&quot;&lt;ref name=&quot;:25&quot; /&gt; (454)

Perhaps not at the same scale as these but as important in 1850s designs was a technology that turned iron into steel, which could then be drawn into fine wires.&lt;ref name=&quot;:3&quot;&gt;{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}&lt;/ref&gt; Steel was refined to a malleable state so that thin blades could be curved into concentric circles (called hoops) and connected with wires to form the cage.

Technological advancements not only influenced the economy but brought a major change in the fashion styles worn by men and women. As the Victorian era was based on the principles of gender, race and class.&lt;ref&gt;{{cite journal|last1=Graham|first1=P|title=The Victorian Era|url=https://archive.org/details/in.ernet.dli.2015.261548|journal=Digital Library of India}}&lt;/ref&gt; Much advancement was in favor of the upper class as they were the ones who could afford the latest technology and change their fashion styles accordingly.  In 1830s there was introduction of horse hair crinoline that became a symbol of status and wealth as only the upper-class women could wear it.  In 1850s there were more fashion technological advancements hence 1850s could rightly be called a revolution in the Victorian fashion industry such as the innovation of artificial cage crinoline that gave women an artificial hourglass silhouette without layers of petticoats, which was lighter and more hygienic.&lt;ref&gt;{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}&lt;/ref&gt; Synthetic dyes, such as [[mauveine]] (aniline purple), were introduced in 1856, adding bright colours to garments. In 1855's ''[[Haute couture]]'' was introduced as tailoring became more mainstream in years to follow.&lt;ref&gt;{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}&lt;/ref&gt;

Charles Frederick Worth, a prominent English designer, became popular amongst the upper class though its city of destiny always is Paris. Haute couture became popular at the same time that sewing machines were invented.&lt;ref name=&quot;Haute Couture&quot;&gt;{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}&lt;/ref&gt; Princess [[Eugénie de Montijo|Eugenie]] of France wore the Englishman dressmaker, Charles Frederick Worth's couture and he instantly became famous in France though he had just arrived in Paris a few years ago. In 1855, Queen Victoria and Prince Albert of Britain welcomed [[Napoleon III]] and Eugenie of France to a full state visit to England. Eugenie was considered a fashion icon in France. Queen Victoria, who had been the fashion icon for European high fashion, was inspired by Eugenie's style and the fashions she wore.{{Citation needed|date=October 2025}} Later, Queen Victoria also appointed Charles Frederick Worth as her dress maker and he became a prominent designer amongst the European upper class. Charles Frederick Worth is known as the father of the haute couture as later the concept of labels were also invented in the late 19th century as custom, made to fit tailoring became mainstream.&lt;ref&gt;{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}&lt;/ref&gt;

By the 1860s, when made-to-fit tailoring was popular in Europe, crinolines were considered impractical. In the 1870s, women preferred more slimmer silhouettes, hence bodices grew longer and the polonaise, a skirt and bodice made together, was introduced. In 1870s the Cuirass Bodice, a piece of armour that covers the torso and functions like a corset, was invented. Towards the end of Victoria's reign, dresses were flared naturally as crinolines were rejected by middle-class women. Designers such as Charles Frederick Worth were also against them. All these inventions and changes in fashion led to women's liberation as tailored looks improved posture and were more practical.&lt;ref name=&quot;Haute Couture&quot;/&gt;

dressmakers, couturiers, modistes

== Home decor ==
{{main|Victorian decorative arts}}

Home decor started spare, veered into the elaborately draped and decorated style we today regard as Victorian, then embraced the retro-chic of [[William Morris]] as well as pseudo-[[Japonaiserie]].

== Myths and Oversimplifications ==

=== Modesty ===
{{main|Victorian morality}}
{{Original research|section|date=May 2008}}
[[File:1868-skirt-lengths-girl-ages-Harpers-Bazar.gif|thumb|upright|&quot;The proper length for little girls' skirts at various ages&quot;, from ''[[Harper's Bazaar]]'', showing a 1900 idea of how the hemline should descend towards the ankle as a girl got older]]Many myths and exaggerations about the period persist to the modern day. Examples include the idea of men's clothing is seen as formal and stiff, women's as elaborate and over-done; clothing covered the entire body, and even the glimpse of an ankle was scandalous. Critics contend that [[corset]]s constricted women's bodies and lives. Homes are described as gloomy, dark, cluttered with massive and over-ornate furniture and proliferating [[bric-a-brac]]. Myth has it that even piano legs were scandalous, and covered with tiny [[pantalette]]s.

=== Tight Lacing ===
Tight-lacing, which was not possible until the development of the grommet in 1828, was famously controversial in the Victorian age, generating many column inches of profitable newspaper copy, in part because it was (and still is) fetishistic and subversive in that adolescent girls used it as a means of rebellion and upper-working- or lower-middle-class shop girls saw it as a means of upward mobility.&lt;ref name=&quot;:21&quot;&gt;{{Cite book|title=Fashion and Fetishism: Corsets, Tight-Lacing and Other Forms of Body-sculpture|last=Kunzle|first=David|publisher=History Press|year=2013|isbn=978 0 7524 9545 3|location=Stroud, Gloucestershire|pages=}}&lt;/ref&gt; (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.&lt;ref name=&quot;:21&quot; /&gt; ()


In truth, men's formal clothing may have been less colourful than it was in the previous century, but brilliant [[waistcoat]]s and [[cummerbund]]s provided a touch of colour, and [[smoking jacket]]s and [[robe|dressing gown]]s were often of rich Oriental [[brocade]]s. This phenomenon was the result of the growing textile manufacturing sector, developing mass production processes, and increasing attempts to market fashion to men.&lt;ref name=&quot;shannon597&quot;/&gt; Corsets stressed a woman's sexuality, exaggerating hips and bust by contrast with a tiny waist. Women's [[evening gown]]s bared the shoulders and the tops of the breasts. The [[jersey dress]]es of the 1880s may have covered the body, but the stretchy novel fabric fit the body like a glove.&lt;ref&gt;{{cite book |last=Gernsheim |first=Alison |title=Victorian &amp; Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}&lt;/ref&gt;

Home furnishing was not necessarily ornate or overstuffed. However, those who could afford lavish draperies and expensive ornaments, and wanted to display their wealth, would often do so. Since the Victorian era was one of increased social mobility, there were ever more ''[[nouveaux riches]]'' making a rich show.

The items used in decoration may also have been darker and heavier than those used today, simply as a matter of practicality. London was noisy and its air was full of [[soot]] from countless coal fires. Hence those who could afford it draped their windows in heavy, sound-muffling curtains, and chose colours that didn't show soot quickly. When all washing was done by hand, curtains were not washed as frequently as they might be today.

There is no actual evidence that piano legs were considered scandalous. Pianos and tables were often draped with [[shawl]]s or cloths—but if the shawls hid anything, it was the cheapness of the furniture. There are references to lower-middle-class families covering up their [[pine]] tables rather than show that they couldn't afford [[mahogany]]. The piano leg story seems to have originated in the 1839 book, ''A Diary in America'' written by Captain [[Frederick Marryat]], as a satirical comment on American prissiness.&lt;ref&gt;{{cite book |last1=Marryat |first1=C.B. |title=A Diary in America: With Remarks on Its Institutions |date=1839 |publisher=Longman, Orme, Brown, Green, and Longmans |location=London, England |volume=2 |pages=246–247 |url=https://books.google.com/books?id=2-VEAAAAIAAJ&amp;pg=PA246}} From pp. 246-247:  &quot;I was requested by a lady to escort her to a seminary for young ladies, and on being ushered into the reception-room, conceive my astonishment at beholding a square piano-forte with four ''limbs''. However, that the ladies who visited their daughters, might feel in its full force the extreme delicacy of the mistress of the establishment, and her care to preserve in their utmost purity the ideas of the young ladies under her charge, she had dressed all these four limbs in modest little trousers, with frills at the bottom of them!&quot;&lt;/ref&gt;

Victorian manners may have been as strict as imagined—on the surface. One simply did not speak publicly about sex, childbirth, and such matters, at least in the respectable middle and upper classes. However, as is well known, discretion covered a multitude of sins. Prostitution flourished. Upper-class men and women indulged in [[adultery|adulterous]] liaisons.

== Gallery ==
{{gallery
|2=A mid-Victorian interior: ''Hide and Seek'' by [[James Tissot]], c. 1877
Image:Winterhalter Elisabeth.jpg|3=Dress designed by [[Charles Frederick Worth]] for [[Elisabeth of Bavaria|Elisabeth of Austria]] painted by [[Franz Xaver Winterhalter]].|4=File:Frith A Private View detail.jpg|5=[[William Powell Frith]]'s painting of 1883 contrasts women's [[Aesthetic dress]] (left and right) with fashionable attire (center).|6=File:Tissot lilacs 1875.jpg|7=Day dress, c. 1875 [[James Tissot]] painting.|8=File:James Abbot McNeill Whistler 011.jpg|9=[[James McNeill Whistler|Whistler]]'s [[Portrait of Lady Meux]], 1882
Image:Jeanna_Samary-Renoir.png|10=[[Pierre-Auguste Renoir|Renoir]]'s portrait of [[Jeanne Samary]] in an [[evening gown]], 1878|11=File:Melville_-_Queen_Victoria.jpg|12=Portrait by [[Alexander Melville (artist)|Alexander Melville]] of [[Victoria of the United Kingdom|Queen Victoria]], 1845|13=File:Henry Treffry Dunn Rossetti and Dunton at 16 Cheyne Walk.jpg|14=An artistic interior: [[Dante Gabriel Rossetti]] reading to [[Theodore Watts-Dunton]] in the drawing room at No. 16 [[Cheyne Walk]], 1882|15=File:Punch - Masculine beauty retouched1.png|16=Men's swimwear: Cartoon from ''[[Punch (magazine)|Punch]]'' by [[George du Maurier]]}}

== See also ==
* [[Emily Clapham]]
* [[Victorian decorative arts]]
* [[Victorian dress reform]]
* [[Victorian morality]]
* [[Victoriana]]
* [[Women in the Victorian Era]]
* [[Charles Frederick Worth]]

=== Time periods ===
* [[1830s in fashion]]
* [[1840s in fashion]]
* [[1850s in fashion]]
* [[1860s in fashion]]
* [[1870s in fashion]]
* [[1880s in fashion]]
* [[1890s in fashion]]

=== Women's clothing ===
* [[Corset]]
* [[Corset controversy]]
* [[Tightlacing]]
* [[Bloomers (clothing)|Bloomers]]
* [[Bodice]]

=== Contemporary interpretations ===
* [[Steampunk]]
* [[Neo-Victorian]]
* [[Lolita Fashion|Lolita]]

== References ==
{{Reflist}}

== Further reading ==
*{{cite book |author=Phipps, Elena| title= ''From Queen to Empress: Victorian dress 1837-1877'' | location=New York | publisher=The Metropolitan Museum of Art | year=1988 | isbn=0870995340| url= http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/69547/rec/235 | display-authors=etal}}
* Sweet, Matthew – ''Inventing the Victorians'', St. Martin's Press, 2001 {{ISBN|0-312-28326-1}}

== External links ==
* [http://www.victorians.co.uk/victorian-fashion Victorian Fashion] {{Webarchive|url=https://web.archive.org/web/20180407223711/http://www.victorians.co.uk/victorian-fashion |date=7 April 2018 }}
* [https://www.victorianvoices.net/topics/fashion/index.shtml VictorianVoices.net] – Fashion articles and illustrations from Victorian periodicals; extensive fashion image gallery
* [http://www.cracked.com/article_19575_5-ridiculous-sex-myths-from-history-you-probably-believe.html Victorian myths]
* [http://www.victorianstation.com/lifestylemenu.htm Victorian fashion, etiquette, and sports] {{Webarchive|url=https://web.archive.org/web/20180103162620/http://www.victorianstation.com/lifestylemenu.htm |date=3 January 2018 }}
* [http://www.thesmartset.com/article/article12180701.aspx Background on &quot;A Diary in America&quot;]
* [http://www.mccord-museum.qc.ca/en/keys/webtours/VQ_P2_17_EN.html Form and Fashion] —  the evolution of women's dress during the 19th century (many photographs)
* [http://www.mccord-museum.qc.ca/en/keys/games/jeu2/ Educational Game: Mix and Match] — build a 19th-century dress using a virtual mannequin
* {{cite web |publisher= [[Victoria and Albert Museum]]
 |url= http://www.vam.ac.uk/content/articles/v/victorian-dress-at-v-and-a/
 |title= Victorian Dress
 |work= Fashion, Jewellery &amp; Accessories
 |date= 14 January 2011
 |access-date= 2011-04-03}}
*[http://cv.vic.gov.au/stories/creative-life/fashion-detective-fashion-fiction-and-forensics/ Fashion detective: Fashion, Fiction and Forensics in nineteenth century Australian fashion] on Culture Victoria

{{Timeline of clothing and fashion|state=collapsed}}{{Victorian era|state=collapsed}}

[[Category:Victorian fashion| ]]
[[Category:19th-century fashion|*]]
[[Category:1900s fashion]]
[[Category:History of Western fashion]]
[[Category:19th century in the arts]]

=From ''Women in the Victorian era''=
===Victorian women's fashion===
{{Multiple issues|{{tone|date=March 2023}}
{{more footnotes needed|date=March 2023}}|section=y}}{{Further|Victorian fashion}}
The ideal Victorian woman was pure, chaste, refined, and modest. This ideal was supported by etiquette and manners. The etiquette extended to the pretension of never acknowledging the use of undergarments (sometimes generically referred to as &quot;unmentionables&quot;). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: &quot;[those] are not things, my dear, that we speak of; indeed, we try not even to think of them&quot;, in contrast to current norms.&lt;ref&gt;{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=20}}&lt;/ref&gt; The pretence of avoiding acknowledgement of anatomical realities met with embarrassing failure on occasion. In 1859, the Hon. Eleanor Stanley wrote about an incident where the [[Louisa Cavendish, Duchess of Devonshire|Duchess of Manchester]] moved too quickly while manoeuvring over a [[stile]], tripping over her large [[hoop skirt]]:

{{blockquote|[the Duchess] caught a hoop of her cage in it and went regularly head over heels lighting on her feet with her cage and whole petticoats above, above her head. They say there was never such a thing seen – and the other ladies hardly knew whether to be thankful or not that a part of her undergarments consisted in a pair of scarlet tartan [[knickerbockers (clothing)|knickerbockers]] (the things Charlie shoots in) which were revealed to the view of all the world in general and the [[Aimable Pélissier|Duc de Malakoff]] in particular&quot;.&lt;ref&gt;{{cite book|last=Cunnington|first=C. Willett|title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations|year=1990|publisher=Dover Publications|isbn=978-0-486-26323-6|pages=20–1}}&lt;/ref&gt;}}

However, despite the fact that Victorians considered the mention of women's undergarments in mixed company unacceptable, men's entertainment made great comedic material out of the topic of ladies' [[bloomers (clothing)|bloomers]], including men's magazines and music hall skits.&lt;ref&gt;{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=22}}&lt;/ref&gt;

Victorian women's clothing followed trends that emphasised elaborate dresses, skirts with wide volume created by the use of layered material such as [[crinoline]]s, hoop skirt frames, and heavy fabrics. Because of the impracticality and health impact of the era's fashions, a [[Victorian dress reform|dress reform movement]] began among women.

The ideal silhouette of the time demanded a narrow waist, which was accomplished by constricting the abdomen with a laced [[corset]]. While the silhouette was striking, and the dresses themselves were often exquisitely detailed creations, the fashions were cumbersome. At best, they restricted women's movements and at worst, they had a harmful effect on women's health. Physicians turned their attention to the use of corsets and determined that they caused several medical problems: compression of the thorax, restricted breathing, organ displacement, poor circulation, and prolapsed uterus.&lt;ref name=&quot;O'Connor&quot;/&gt;

Articles advocating the reform of women's clothing by the British National Health Society, the Ladies' Dress Association, and the [[Rational Dress Society]] were reprinted in ''The Canada Lancet'', Canada's medical journal. In 1884, Dr J. Algernon Temple of Toronto even voiced concern that the fashions were having a negative impact on the health of young women from the working classes. He pointed out that a young working-class woman was likely to spend a large part of her earnings on fine hats and shawls, while &quot;her feet are improperly protected, and she wears no flannel petticoat or woollen stockings&quot;.&lt;ref name=&quot;O'Connor&quot;/&gt;
[[File:Bloomers.jpg|thumb|1850s illustration of a woman wearing [[bloomers]]]]
[[Florence Pomeroy]], Lady Haberton, was president of the Rational Dress movement in Britain. At a National Health Society exhibition held in 1882, Viscountess Haliburton presented her invention of a &quot;[[divided skirt]]&quot;, which was a long skirt that cleared the ground, with separate halves at the bottom made with material attached to the bottom of the skirt. She hoped that her invention would become popular by supporting women's freedom of physical movement, but the British public was not impressed by the invention, perhaps because of the negative &quot;unwomanly&quot; association of the style with the American [[Bloomers]] movement.&lt;ref&gt;{{cite book|last=Murray|first=Janet Horowitz|title=Strong-Minded Women and Other Lost Voices from 19th Century England|year=1982|publisher=Pantheon Books|location=New York|isbn=0-394-71044-4|pages=[https://archive.org/details/strongmindedwome00jane/page/68 68–70]|url=https://archive.org/details/strongmindedwome00jane/page/68}}&lt;/ref&gt; [[Amelia Jenks Bloomer]] had encouraged the wearing of visible bloomers by feminists to assert their right to wear comfortable and practical clothing, but it was no more than a passing fashion itself among radical feminists. The movement to reform women's dress would persist and have long-term success, however; by the 1920s, [[Coco Chanel]] was successful at selling a progressive, far less restrictive silhouette that abandoned the corset and raised hemlines. The new silhouette symbolised modernism for trendy young women and became the 20th century standard. Other Paris designers continued reintroducing pants for women and the trend was gradually adopted over the next century.

Fashion trends, in one sense, travelled &quot;full circle&quot; over the course of the Victorian era. The popular women's styles during the [[Georgian era]], and at the very beginning of Victoria's reign, emphasized a simple style influenced by flowing gowns worn by women in [[Ancient Greek clothing|Ancient Greece]] and [[Clothing in ancient Rome|Rome]]. The [[Empire waist]] silhouette was replaced by a trend towards ornate styles and an artificial silhouette, with the restrictiveness of women's clothing reaching its low point during the mid-century passion for narrow corseted waists and hoop skirts. The iconic wide-brimmed women's hats of the later Victorian era also followed the trend towards ostentatious display. Hats began the Victorian era as simple [[Bonnet (headgear)|bonnets]]. By the 1880s, milliners were tested by the competition among women to top their outfits with the most creative (and extravagant) hats, designed with expensive materials such as silk flowers and exotic plumes such as ostrich and peacock. As the Victorian era drew to a close, however, fashions were showing indications of a popular backlash against excessive styles. Model, actress and socialite [[Lillie Langtry]] took London by storm in the 1870s, attracting notice for wearing simple black dresses to social events. Combined with her natural beauty, the style appeared dramatic. Fashions followed her example (as well as Queen Victoria's wearing of mourning black later in her reign). According to [[Harold Koda]], the former Curator-in-chief of the [[Costume Institute at The Met|Metropolitan Museum of Art's Costume Institute]],&lt;ref&gt;{{cite web|url=http://www.metmuseum.org/about-the-museum/press-room/exhibitions/2014/death-becomes-her|title=Death Becomes Her: A Century of Mourning Attire : October 21, 2014-February 1, 2015|website=Metmuseuim.org|access-date=7 November 2021}}&lt;/ref&gt; &quot;The predominantly black palette of [[mourning]] dramatizes the evolution of period silhouettes and the increasing absorption of fashion ideals into this most codified of etiquettes,&quot; said Koda, &quot;The veiled widow could elicit sympathy as well as predatory male advances. As a woman of sexual experience without marital constraints, she was often imagined as a potential threat to the social order.&quot;

====Evolution of Victorian women's fashion====
&lt;gallery&gt;
File:Fashion plate December 1844.jpg|Ladies' December Fashions (1844). Hand-coloured steel engraving from a women's magazine.
File:Thegalleryofhmscalcutta james tissot 1876.jpg|''[[The Gallery of HMS Calcutta]]'' by [[James Tissot]] (1876). [[Bustle]]s were fashionable in the 1870s and 1880s.
File:Mrs lillie langtry george frederic watts 1880.jpg|''Mrs. Lillie Langtry'' by [[George Frederic Watts]] (1880).
File:Five-women-on-queenslander-steps-r.jpg|Fashionable women in [[Queensland]], Australia around 1900.
&lt;/gallery&gt;


{{Short description|Irish writer (born 1963)}}
{{Use Irish English|date=August 2025}}
{{Use dmy dates|date=August 2025}}
{{Infobox writer
| name               = Darach Ó Scolaí
| image              = Darach Ó Scolaí.JPG
| alt                = Man holding prize-winning book
| caption            = Ó Scolaí in 2019
| birth_name         = Darach Ó Scolaí
| birth_date         = {{Birth date and age|1963|df=y}}
| birth_place        = [[County Galway]], The Republic of Ireland
| death_date         = 
| death_place        = 
| occupation         = Writer, artist, publisher
| alma_mater         = [[University of Galway]]
| years_active       = 1998–present
| genre            = Novel, retelling, translation, play, screenplay, illustrated book for children and adults
| other_names        = 
| spouse             = 
| children           = 3
| awards             = [[Awards and Honors received by Darach Ó Scolaí|Awards and Honors]]
| signature          = 
| website            = 
}}[[File:Darach Ó Scolaí.JPG|thumb|Darach Ó Scolaí, holding ''Oileán an Órchiste'' (his translation of Robert Louis Stevenson's ''Treasure Island'')]]


== Darach Ó Scolaí ==
Darach Ó Scolaí (&lt;small&gt;Irish:&lt;/small&gt; [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963&lt;ref&gt;{{Cite web|url=https://portraidi.ie/en/darach-o-scolai/|title=Darach Ó Scolaí|date=20 October 2017|website=Portráidí (Portraits of Irish-Language Writers)|access-date=1 August 2025}}&lt;/ref&gt;) is an Irish author who works in a number of genres, from novels, plays and screenplays to illustrated books for children and adults. He began his literary career in 1998 writing screenplays, stage plays, retellings and translations; he began to publish novels in 2008. Ó Scolaí is widely recognized as a leading figure in contemporary Irish literature, known as “one of the most important Irish language writers of his generation”&lt;ref&gt;{{Cite journal|last=Poirtéir|first=Cathal|date=2022|title=? Suil an Daill: Constant Tensions and Shifting Allegiances|url=https://booksirelandmagazine.com/suil-an-daill-constant-tensions-and-shifting-allegiances/|journal=Books Ireland}}&lt;/ref&gt; and &quot;one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge].&quot;&lt;ref name=&quot;:17&quot; /&gt; His writing has been called “the high literature of the Irish language.”&lt;ref&gt;Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in &quot;Táin Bó Cuailnge.&quot; ''Leabhar Breac''. Retrieved 25 August 2025.&lt;/ref&gt;

Much of his fiction is based on a knowledge of traditional Irish tales and narrative practices as well as Irish history. He specializes in literary and [[wikipedia:Historical_fiction|historical fiction]], or as novelist Alan Titley says, Ó Scolaí’s “peak (for now), or at least his greatest imaginative interest, is the historical novel [tá an chuma air gurb é a bhuaic (go fóill), nó ar a laghad, a mhórspéis samhlaíochta, an t-úrscéal staire].”&lt;ref name=&quot;:11&quot;&gt;{{Cite journal|last=Titley|first=Alan|date=Fall 2020|title=An Stíl Go Deo!: Soather Dharach Uí Scolaí (The style would be forever!: Worker Darach Ó Scolaí)|url=https://www.jstor.org/stable/27046090|journal=Comhar|volume=80, No. 10|pages=27|via=JSTOR}}&lt;/ref&gt; His retellings of old stories and tales from their original Middle and Early-Modern Irish into Modern Irish ([[wikipedia:Irish_language|Gaeilge]]) are respected for their accessibility to students and language learners as well as for their artistry.

Ó Scolaí also regularly reviews books and lectures and writes on literature and culture. 

Beyond his writing, Ó Scolaí is a publisher and has co-produced a number of film, television shows and stage plays. 

== Life ==
Ó Scolaí was born in Dublin and raised in the Galway [[wikipedia:Gaeltacht#Galway Gaeltacht|Gaeltacht]] (Irish-speaking) regions of Cois Fharraige on the north shore of Galway Bay, in the Republic of Ireland, where he lives now with his wife and children in Lochán Beag (Indreabhán).&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|date=30 October 2024|title=Duais don úrscéal liteartha is fearr buaite ag Darach Ó Scolaí ag Oireachtas na Samhna|url=https://tuairisc.ie/duais-don-ursceal-liteartha-is-fearr-buaite-ag-darach-o-scolai-ag-oireachtas-na-samhna/|journal=Tuairisc}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Ní Scolaí|first=Aifric|date=2024|title=Darach Ó Scolaí|url=https://www.taiscecf.ie/ealaiontoiri?category=Scr%C3%ADbhneoir|journal=Taisce Chois Fharraige}}&lt;/ref&gt; 

He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.&lt;ref&gt;{{Cite web|url=https://www.linkedin.com/in/darach-ó-scolaí-20026920/|title=Darach Ó Scolaí|last=Ó Scolaí|first=Darach|date=August 2025|website=LinkedIn}}&lt;/ref&gt; 

=== Writing and Publishing ===
Ó Scolaí writes in Irish ([[wikipedia:Irish_language|Gaeilge]]), his native language, and lives in an area defined for the predominant presence of Irish as the vernacular language, the language spoken at home. Irish was the language of his parents' home and is the language of children as well. He is fluent in Irish and English and conversant in French. 

None of his works has been translated into English. 

==== Leabhar Breac ====
In 1995 Darach Ó Scolaí and his brother Caomhán Ó Scolaí — a [[wikipedia:Typography|typographer]] and designer — founded the publishing house Leabhar Breac at Indreabhán (Inverin), County Galway. Their father “Séamas Ó Scolaí was an editor at An Gúm and worked on the Irish-English dictionary team [bhí a n-athair Séamas Ó Scolaí ina eagarthóir sa Ghúm agus d’oibrigh sé ar fhoireann an fhoclóra Gaeilge-Béarla].”&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}&lt;/ref&gt; Darach Ó Scolaí has been publisher and literary editor at Leabhar Breac since its founding. 

Named for [[wikipedia:An_Leabhar_Breac|An Leabhar Breac (The Speckled Book)]], Leabhar Breac publishing house has more than 140 books in print.&lt;ref name=&quot;:0&quot; /&gt; Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”&lt;ref name=&quot;:0&quot; /&gt; Besides the content, Leabhar Breac is known for the typically &quot;superb [thar cionn]&quot; quality of the design and production of the &quot;physical book [leabhar fisiciúil].&quot;&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Ní Mhuilneoir|first=Gráinne|date=30 July 2024|title=‘Bláthnaid’ – leabhar álainn i sraithín álainn faoi mhná|url=https://tuairisc.ie/blathnaid-leabhar-alainn-i-sraithin-alainn-faoi-mhna/|journal=Tuairisc}}&lt;/ref&gt; Its books regularly win awards for literary and artistic quality. Leabhar Breac also publishes translations for children and adults from various early versions of Irish as well as from French and English (and has published translations of books for young readers from Spanish, Catalan, and Italian as well).

Leabhar Breac prints its books in Ireland.

=== Stage and Screen ===

==== Rosg ====
In 1998 along with Ciarán Ó Cofaigh,&lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=http://www.rosg.ie/en/about/History_6/|title=About Us: History|date=July 2025|website=Rosg|access-date=1 August 2025}}&lt;/ref&gt; Ó Scolaí co-founded the film and television production company [http://www.rosg.ie/en/ Rosg] and was co-director until 2006. Rosg produced Ó ScolaÍ’s films ''Cosa Nite'' (1999), ''An Leabhar'' (2001) and ''Na Cloigne'' (2010). He left Rosg in 2006 to devote his time to other artistic activities. 

==== Ealaín ar Oileán ====
In 2004, along with Val Balance, Ó Scolaí co-founded the annual artists' symposium Ealaín ar Oileán (trans., Art on an Island). The Irish-language symposium was held annually in the Áras Éanna arts and cultural center on Inis Oírr ([[wikipedia:Inisheer|Inisheer]], the smallest of the [[wikipedia:Aran_Islands|Aran Islands]]) from 2004 to 2013. Ó Scolaí was its co-director from its founding&lt;ref&gt;{{Cite web|url=https://ga.wikipedia.org/wiki/Darach_Ó_Scolaí.|title=Darach Ó Scolaí|date=3 February 2024|website=Vicipéid|access-date=1 July 2025}}&lt;/ref&gt; until 2013.

Besides being its co-director, Ó Scolaí has taken part in this conference as an artist&lt;ref&gt;{{Cite journal|date=16 January 2005|title=Darach Ó Scolaí|url=https://web.archive.org/web/20050116163252/http://bliainiris.com/authors/darach_oscolai.html|journal=Bliainiris}}&lt;/ref&gt; and writer&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=http://ealainaroilean.ie/ealainaroilean.html|title=The Conference|date=7 September 2013|website=Ealaín ar Oileán|archive-url=https://web.archive.org/web/20130907083744/http://ealainaroilean.ie/ealainaroilean.html|archive-date=7 September 2013|access-date=1 August 2025}}&lt;/ref&gt;.

==== Salamandar ====
In 2006 Ó Scolaí founded the stage production company Salamandar and directed his own play ''An Braon Aníos''. His plays ''An tSeanbhróg'' (2009) and ''Craos'' (2008) were also produced by Salamandar.&lt;ref name=&quot;:19&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/product-category/darach-o-scolai/|title=Darach Ó Scolaí|date=2024|website=Leabhar Breac|access-date=1 July 2025}}&lt;/ref&gt; 

== Works ==

=== Novels ===
* [[wikipedia:An_Cléireach|''An Cléireach'' (trans., ''The Clerk'')]], Leabhar Breac, 2007. The Oireachtas Prize for Literary Fiction, 2007; The Ó Súilleabháin Award (Book of the Year) in 2008, and &quot;named as ‘the best novel since the turn of the Century’ by Comhar.&quot;&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-cleireach/|title=An Cléireach - Leabhar Breac - Irish language novel|website=Leabhar Breac|language=en-US|access-date=2025-10-24}}&lt;/ref&gt;
* ''Na Comharthaí'' (trans., ''The Signs''), Leabhar Breac, 2014.
* ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. The Oireachtas Prize for Literary Fiction, 2019.&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/suil-an-daill/|title=Súil an Daill|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024&lt;ref name=&quot;:7&quot; /&gt;; The Ó Súilleabháin Award (Book of the Year) in 2025; featured in the 2025 Listen-Up Irish Summer Challenge for students of the Irish language.&lt;ref&gt;{{Cite news|url=https://connachttribune.ie/novel-approach-helps-people-learn-irish-in-a-creative-way/|title=Novel approach helps people learn Irish in a creative way|last=Murphy|first=Judy|date=3 October 2025|work=Connaught Tribune|access-date=24 October 2025}}&lt;/ref&gt;

=== Retellings, Translations and Editions ===
The retellings and translations are into modern Irish.

* ''Feis Tigh Chonáin'' (trans., ''The Feast of Conán's House''), Leabhar Breac, 2000; a retelling of a 15&lt;sup&gt;th&lt;/sup&gt;-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/feis-tigh-chonain/|title=Feis Tigh Chonáin|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-ceithearnach-caolriabhach/|title=An Ceithearnach Caolriabhach|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), Leabhar Breac, 2017, both a modern edition of an 11th-century epic and an annotated edition.&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/tain-bo-cuailnge-2-2/|title=Táin Bó Cuailnge|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;  &quot;''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017.&quot;&lt;ref name=&quot;:19&quot; /&gt;
* ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; Part of the [[wikipedia:Ulster_Cycle|Ulster Cycle]], ''Deirdre'' is a retelling  of the story of possibly the most widely known Irish figure from the early tales and sagas.&lt;ref&gt;{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}&lt;/ref&gt;
* ''Bláthnaid'', Leabhar Breac, 2024, a “picture book for adults” with artist Anastasia Melnykova; “one of the great stories of the [[wikipedia:Ulster_Cycle|Ulster Cycle]].”&lt;ref name=&quot;:4&quot; /&gt;
* ''Sadhbh,'' Leabhar Breac, 2025, a picture book for adult readers, illustrated by Alé Mercado; a retelling of the medieval tale ''Ceasacht Inghine Ghuile (''trans., ''The Complaint of Guile's Daughter'').&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado&lt;ref name=&quot;:5&quot; /&gt;; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''&lt;ref name=&quot;:5&quot; /&gt;

=== For Young Readers ===
Ó Scolaí has written illustrated books for young readers (8–10 years old) in two series, the Fionn Series and the Scéalta Staire series, and translated a large number of classics and popular books for children of all ages. The number of these written and translated works suggests a commitment to children and their literacy in Irish. 

The Fionn Series “is a retelling ... of the great legends of the Fianna for the young Irish readers of today.”&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/doiteoir-na-samhna/|title=Dóiteoir na Samhna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; [[wikipedia:The_Boyhood_Deeds_of_Fionn|Macgnímartha Finn (The Boyhood Deeds of Fionn)]] is a medieval story in the [[wikipedia:Fenian_Cycle|Fenian Cycle]].

* ''An Bradán Feasa'' (trans., ''The Salmon of Knowledge''), Leabhar Breac, 2010, “shortlisted for the Réics Carlo award 2010.”&lt;ref name=&quot;:9&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/an-bradan-feasa/|title=An Bradán Feasa|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.&lt;ref name=&quot;:6&quot; /&gt;
* ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/|title=Bodach an Chóta Lachna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;

The Scéalta Staire (Historical Stories) series&lt;ref name=&quot;:9&quot; /&gt;

* ''Mánas Ó Dónaill'', 2000.
* ''Seán Ó Néill'', Leabhar Breac, 2000.
* ''Gráinne Mhaol Ní Mháille'', Leabhar Breac, 2003.
* ''Tadhg Dall Ó hUiginn'', Leabhar Breac, 2003.

==== Translations ====

* Robert Louis Stevenson, ''Oileán an Órchiste'' (trans. of ''Treasure Island''), Leabhar Breac, 2014.&lt;ref&gt;{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&amp;oldid=1296394643|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Robert Louis Stevenson, ''An Fuadach'' (trans. of ''Kidnapped''), Leabhar Breac, 2016.
* Clement Clarke Moore, ''Cuairt San Nioclás'' (trans. of ''A Visit from St. Nicholas'', or &quot;'Twas the Night Before Christmas&quot;), Leabhar Breac, 2022.
'''''The Corto Maltese Graphic Novels'''''

Written in Italian by Hugo Pratt and translated by Ó Scolaí, both adults and teenagers read this series of Italian adventure graphic novels.&lt;ref&gt;{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&amp;oldid=1298285365|journal=Wikipedia|language=en}}&lt;/ref&gt; Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among &quot;The 30 Irish books that Irish people love.&quot;&lt;ref&gt;{{Cite journal|last=Ó Murchú|first=Eoin P.|date=09/06/2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil [The 30 Irish books that Irish people love]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}&lt;/ref&gt;

* Hugo Pratt, ''Corto: Port na Farraige Goirt'', Leabhar Breac, 2013.
* Hugo Pratt, ''Corto: The Golden House in Samarkand'', 2014.
* Hugo Pratt, ''Corto: Na Liopard-Fhir ó Rufiji'' (trans. of ''Corto: The Leopard Men of Rufiji''), Leabhar Breac, 2014.
* Hugo Pratt, ''Corto: In Ainm Dé Uilthrócairigh'' (trans. of ''Corto: In the Name of God All-Merciful''), Leabhar Breac, 2014.
* Hugo Pratt, ''Corto: Tóraíocht Eile'' (trans. of ''Corto: Another Quest''), Leabhar Breac, 2014.
* Hugo Pratt, ''Corto: Sa tSibéir'' (trans. of ''Corto: In Siberia''), Leabhar Breac, 2016.
'''''Other Translations for Children'''''

Ó Scolaí has translated into Irish six books from the ''Le Pavillon Noir'' (trans., ''Jolly Roger'') series by Alain Surget; four books from the ''Catalan First Steps'' series by Enric Lluch Girbés and the ''Caitlín &amp; Cormac'' series by Joan Carles; three books from the ''Louisette le Taupe'' series by Bruno Heitz, and three books from the ''Loup'' series by Orianne Lallemand.

=== Plays and Screenplays ===

==== Stage Plays ====
Ó Scolaí was writer and director of the original productions of two plays in the ''Trí Bhraon'' (trans., ''Three Drops'') trilogy; ''Coinneáil Orainn'' was directed by Darach Mac Con Iomaire and staged by An Taibhdhearc. All three plays have been published in book form by Leabhar Breac.

* ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32553|title=Coinneáil Orainn|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}&lt;/ref&gt; The first play in the ''Trí Bhraon'' (''Three Drops'') trilogy. [[wikipedia:Taibhdhearc_na_Gaillimhe|An Taibhdhearc]], the national Irish-language theatre of Ireland, toured the country in 2005 with ''Coinneáil Orainn''.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/coinneail-orainn/|title=Coinneáil Orainn|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/person/darach-scola|title=Darach Ó Scolaí|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}&lt;/ref&gt;
* ''Branwen'', 2006, by Darach Ó Scolaí and Ifor ap Glyn, in Irish, Welsh and English, co-produced by Project Arts Centre and Llwyfan Gogledd Cymru, toured the Republic of Ireland and Wales.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32418|title=Branwen|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt;
* ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32461|title=An Braon Aníos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt; The second play in the ''Trí Bhraon'' (''Three Drops'') trilogy. “The Salamandar company toured the country in 2006-07 with this play, and Salamandar also produced a radio version of the play for RTÉ Raidió na Gaeltachta in 2009.”&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/an-braon-anios/|title=An Braon Aníos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.&lt;ref name=&quot;:13&quot;&gt;{{Cite web|url=https://irishplayography.com/play?playid=32867|title=Craos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt; The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.&lt;ref name=&quot;:13&quot; /&gt; A review of the 2008 Salamander performance in the ''Irish Times'' says, “a humorous play which offers plenty to think about, fine acting, and sparklingly witty dialogue.”&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/craos-2/|title=Craos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, &quot;dance and music drama,&quot; co-produced by Ciotóg and Salamandar.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32962|title=A+E|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt;
* ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=33042|title=An tSeanbhróg|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt; and staged in the Axis Arts Centre, Dublin, and the Letterkenny Arts Centre.
* '''In ''Mhuir Fhíondorcha/The Wine-Dark Sea: The Homer Project'', Ó Scolaí's translation of Homer's Cyclops story, performed at the 2019 IMRAM festival'''.&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/culture/books/imram-a-festival-celebrating-the-irish-language-1.4047610|title=Imram: a festival celebrating the Irish language. Liam Carson reveals the myths and legends appearing in this year’s programme|last=Carson|first=Liam|date=11 October 2019|work=The Irish Times|access-date=15 October 2025}}&lt;/ref&gt;

==== Screenplays ====

* ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); &quot;a prose version of ''Cosa Nite'' was published (Rosg 2000).&quot;&lt;ref name=&quot;:9&quot; /&gt; Nominated for an Irish Film and Television Award.&lt;ref&gt;{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}&lt;/ref&gt;
* ''Na Glúnta'' (trans., ''The Generations''), 2001&lt;ref&gt;{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&amp;aid=70&amp;rid=3917&amp;tpl=filmography_dets&amp;only=1&amp;force=1|title=Na Glúnta {{!}} The Irish Film &amp; Television Network|website=www.iftn.ie|access-date=2025-08-25}}&lt;/ref&gt;, co-directors Ciarán Ó Cofaigh &amp; Darach Ó Scolaí, prod. Ciarán Ó Cofaigh, Rosg.
* ''An Leabhar'' (trans., ''The Book''), short film, 2000, (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg) Rosg, 2000.&lt;ref&gt;{{Citation|title=An Leabhar|url=https://www.imdb.com/title/tt0963767/|publisher=Bord Scannán na hÉireann / The Irish Film Board, ROSG|accessdate=2025-08-25|first=Robert|last=Quinn|others=Colm O&amp;apos;Maonlai, Peadar O&amp;apos;Treasaigh, Diarmuid Mac an Adhastair}}&lt;/ref&gt;
* ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.&lt;ref&gt;{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}&lt;/ref&gt;
=== Nonfiction ===
Ó Scolaí's essays and lectures are published and his interviews are broadcast regularly, making for a large body of nonfiction critical and analytical work. Here are a few, almost all published in [https://comhar.ie/iris/scribhneoiri/darach-o-scolai/ Comhar]:

* “Ceol Ciúin na nÉagmaise” (trans., “The Silent Music of Absence ['''the Fall?''']”), an essay on the 2014 Nobel Prize winner for literature, [[wikipedia:Patrick_Modiano|Patrick Modiano]], ''Comhar'', December 2014.
* The Ó Cadhain Lecture: [https://leachtaiuichadhain.clo.ie/leachtai/2014 “Cuimhne agus Díchuimhne (trans., “Memory &amp; Forgetfulness&quot;]), 2014.
* “Rithim agus Réim” (&quot;Rhythm and Register&quot;), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., &quot;From Talk to Twitter&quot;), 2014.
* Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''.
* “Na Geilt i mBun an Tí” (trans., &quot;The Madmen in Charge&quot;), a talk at the Merriman Winter School, Comhar April 2012.&lt;ref name=&quot;:18&quot;&gt;{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}&lt;/ref&gt;
* The EFACIS podcast: Síle Ní Choincheannain talks to Darach Ó Scolaí about the historical novel.

== Critical Reception ==
Ó Scolaí’s style has been called “crisp and elegant, and rich in language while being highly readable,”&lt;ref&gt;{{Cite journal|last=Heussaf|first=Anna|date=Summer 2025|title=Bláthnaid—A tale of love, violence and sorcery retold for readers today|url=https://booksirelandmagazine.com/blathnaid-a-tale-of-love-violence-and-sorcery/|journal=Books Ireland}}&lt;/ref&gt; with “an unsurpassed richness and precision of language.”&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}&lt;/ref&gt; “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.&lt;ref name=&quot;:20&quot; /&gt;   

=== Original Works ===
Ó Scolaí’s first novel, the 2007 ''An Cléireach'' (''The Clerk'') won two prizes and was described as “one of the great historical novels in the Irish language and among the best books written in the language since the beginning of this century.”&lt;ref name=&quot;:12&quot; /&gt; Novelist Alan Titley says, “In ''An Cléireach'' Ó Scolaí creates the Ireland of war in the 17th century more fully than any other Irish writer on the subject of war since ''L’Attaque'' Eoghain Ó Thuairisc around 1798 [In ''An Cléireach'' cruthaíonn Ó Scolaí Éire an chogaidh san 17ú haois níos iomláine ná mar a dhein aon scríbhneoir Gaeilge eile ar ábhar cogaidh ó ''L’Attaque'' Eoghain Uí Thuairisc timpeall ar 1798].”&lt;ref name=&quot;:11&quot; /&gt;{{rp|25, Col. 1a}} Not all the reviews of this first novel were so positive, however; Proinsias O' Drisceoil says for the Irish Times says,&lt;blockquote&gt;This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/news/a-disaffected-clerk-in-the-confederates-1.943070|title=A disaffected clerk in the confederates|last=O' Drisceoil|first=Proinsias|date=5 July 2008|work=The Irish Times|access-date=16 October 2025}}&lt;/ref&gt;&lt;/blockquote&gt; 

In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': &lt;blockquote&gt;In ''An Cléireach'', Ó Scolaí revisits the trauma of Cromwellian Ireland. The primary narrative device is once again the first-hand account, in this case by Tadhg Ó Dúbháin, a clerk and quartermaster in the Confederate Army in 1650. We sample the hardships, the friendships, the tensions, the rivalries, and the petty jealousies amongst comrades in arms, including remnants of the Gaelic literary class, as the Confederate soldiers, increasingly a rabble more than a cohesive unit, retreat in advance of Cromwell’s forces. ''An Cléireach'' concludes with the narrator and his family in exile in continental Europe. But along the retreat route, and central to the novel, members of the Confederate army camp, rest up, and tell versions of a story about the keeper of the treasured manuscript &quot;Saltair an Easpaig&quot; (The Bishop’s Psalter). Their versions raise issues about memory construction, the limitations of individual perspectives, personal agendas, and how minor changes in the telling of a story can alter our understanding of history, Thus, ''An Cléireach'' complements ''Fontenoy'' in moving beyond more realistic recreation of a historical event or period to interrogate the notion of history as construct.&lt;ref&gt;{{Cite book|title=The Oxford Handbook of Modern Irish Fiction|last=Ó Siadhail|first=Pádraig|publisher=Oxford University Press|year=2020|isbn=9780198754893|editor-last=Harte|editor-first=Liam|pages=598–99|chapter=Contemporary Irish Fiction}}&lt;/ref&gt; &lt;/blockquote&gt;

Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, &quot;The book deserves a significant place among the collection of high-quality books published in recent years that would make you feel sorry for someone who does not speak Irish [Tá áit shuntasach ag dul don leabhar i measc an chnuasaigh leabhair ar ardchaighdeán a foilsíodh le roinnt blianta anuas a d’fhágfadh trua agat don té atá gan Ghaeilge].&quot;&lt;ref&gt;{{Cite journal|last=Seoighe|first=Cathal|date=09/26/2022|title=‘Dar leathmhagairle an diabhail, is leabhar den scoth é seo!’ ['According to the devil’s half-wit, this is a great book!’]|url=https://nos.ie/cultur/leabhair/dar-leathmhagairle-an-diabhail-is-leabhar-den-scoth-e-seo/|journal=Nós}}&lt;/ref&gt;

''Bódléar'', Ó Scolaí's most recent book, is a “beautiful novel. There is magic and craftsmanship in it. A small miracle of a book and it is highly recommended.”&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/bodlear-mioruilt-bheag-de-leabhar/|title=Bódléar: Míorúilt bheag de leabhar (Bódléar: A Small Miracle of a Book)|last=Ní Ghairbhí|first=Róisín|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; Éilis Ní Dhuibhne in the ''Irish Times'' says,&lt;blockquote&gt;what a gem! An affectionately gentle satire of the Irish poetic scene during one creatively fluid 19th-century year, the story focuses on a Maigue poet and schoolteacher who goes on a trip to France and returns with camembert, a cafetiere, ‘Fleurs du Mal’, and a mission to convert the local traditionalists to la modernité. Whimsical, hilarious, and subtly learned, it’s absolutely delightful!&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Ní Dhuibhne|first=Éilis|date=30 June 2025|title=Éilís Ní Dhuibhne on the best Irish language books of 2025 so far:
Including a history of the Gaeltacht Civil Rights Movements, a gem of a novel by Darach Ó Scolaí and Joe McHugh’s entertaining account of learning Irish|url=https://www.irishtimes.com/culture/books/review/2025/06/30/eilis-ni-dhuibhne-on-the-best-irish-language-books-of-2025-so-far/|journal=The Irish Times|pages=22}}&lt;/ref&gt;&lt;/blockquote&gt;

=== Retellings and Translations ===

==== ''Táin Bó Cuailnge'' ====
''Táin Bó Cuailnge'' [''The Cattle Raid of Cooley''] is a modern edition of an 11th-century epic into modern Irish.&lt;ref name=&quot;:3&quot; /&gt; Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':&lt;blockquote&gt;Darach Ó Scolaí has ​​achieved a feat in this challenging reworking. He has found a high level of the Irish language to tell his story – as he has done before in his groundbreaking novel An Cléireach (2007, Leabhar Breac) and in his other prose works. This book is a decoration of the language, literature and culture of the Irish language, following the path of the old storytellers and writers and presenting material from the tradition to his own generation according to the understandings of his own time. The book will be a classic that will be of great interest to all readers of the Irish language, both ordinary readers, students, scholars and writers, and there should be a copy in every home in the country. [Tá éacht déanta ag Darach Ó Scolaí san athleagan dúshlánach seo. Tá réim ard den teanga Ghaeilge aimsithe aige lena scéal a inseacht – mar a rinne sé cheana ina úrscéal ceannródaíoch An Cléireach (2007, Leabhar Breac) agus i saothair eile phróis dá chuid. Is maisiú ar an teanga agus ar litríocht agus cultúr na Gaeilge an leabhar seo a leanas conair na seanscéalaithe agus na seanscríobhaithe agus ábhar de chuid an traidisiúin á chur i láthair a ghlúine féin aige de réir thuiscintí a linne féin. Clasaic a bheas sa leabhar a gcuirfidh léitheoirí uilig na Gaeilge, idir ghnáthléitheoirí, mhic léinn, scoláirí agus scríbhneoirí spéis thar na bearta ann, agus ba cheart cóip a bheith i chuile theach sa tír.]&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|last=Denvir|first=Gearóid|date=April 2018|title=Táin Bó Cuailgne|url=https://comhar.ie/iris/78/4/leirmheas/|journal=Comhar|via=JSTOR}}&lt;/ref&gt; &lt;/blockquote&gt;Cathal Poirtéir says, &quot;The freshness and richness of Ó Scolaí’s version are a joy …. The author delights us with the linguistic and stylistic richness of the ancient epic in a modern-Irish version that reflects the original’s spirit and language.&quot;&lt;ref&gt;{{Cite journal|last=Poirtéir|first=Cathal|date=May/June 2018|title=Leabhair Idir Lámha|url=https://www.jstor.org/stable/26564180|journal=Books Ireland|pages=46–47|via=JSTOR}}&lt;/ref&gt;{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's &quot;a wonderful gutsy telling&quot; of ''Táin Bó Cuailnge''.&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/culture/2023/03/11/the-tain-retold-maeve-and-ailills-spat-could-be-out-of-a-soap-opera/|title=The Táin retold: ‘Maeve and Ailill’s spat could be out of a soap opera’|last=Titley|first=Alan|date=11 March 2023|work=The Irish Times|access-date=16 October 2025}}&lt;/ref&gt;

==== ''Deirdre'' ====

Marie Whelton, in &quot;Léann Teanga&quot; (&quot;Language Studies&quot;), in the 2024 ''An Reiviú'' says,&lt;blockquote&gt;this version [of ''Deirdre''] by Darach Ó Scolaí succeeds in skillfully capturing and portraying the complexity of gender and power issues in the ‘Deirdre’ tradition [éiríonn leis an leagan seo le Darach Ó Scolaí castacht cheisteanna na hinscne agus na cumhachta i dtraidisiún scéal Dheirdre a ghabháil agus a léiriú go sciliúil]. … There is no doubt that this new version greatly contributes to the legacy of the story and that it revives that legacy thoughtfully and artistically [Níl amhras faoi ach go gcuireann an leagan úr seo go mór le hoidhreacht an scéil agus go ndéanann sé an oidhreacht sin a athbheochan go tuisceanach agus go healaíonta.].&lt;ref name=&quot;:16&quot;&gt;{{Cite web|url=https://www.tara.tcd.ie/tara8/server/api/core/bitstreams/3c20175a-7631-44b2-8b0f-f454edd712b4/content|title=An Artistic Retelling of Deirdre's Tale and the Defeat of Conor Review of Deirdre or the Ship of Mac Uisnigh by Darach Ó Scolaí [Athinsint Ealaíonta ar Oidhe Dheirdre agus ar Ansmacht Chonchúir Léirmheas ar Deirdre nó Loingeas Mhac Uisnigh le Darach Ó Scolaí]|last=Whelton|first=Marie|date=2024|website=The Review [An Reiviú], Language Studies [Léann Teanga]|access-date=25 September 2025}}&lt;/ref&gt;&lt;/blockquote&gt;
=== Works for Young Readers ===
Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was &quot;Beautiful Irish, but easy to understand for young readers.&quot;&lt;ref&gt;Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in &quot;Bodach an Chóta Lachna&quot; https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.&lt;/ref&gt;

== Awards and Honors ==
Ó Scolaí's works are regularly nominated and make the short list for prizes, an honor in itself, but they are generally not listed here unless they are named as the first-place winner in their category.

=== Oireachtas Prize ===
The Oireachtas Prize is the literary prize awarded by [[wikipedia:Oireachtas_na_Gaeilge|Oireachtas na Gaeilge]], the annual arts festival dedicated to Irish language, arts and culture. Darach Ó Scolaí has won the Oireachtas Prize for Literary Fiction three times, once for ''An Cléireach'' (''The Clerk'') in 2007, for ''Súil an Daill'' (''The Eye of the Blind'') in 2021 and for ''Bódléar'' in 2024.

* 2007, for ''An Cléireach'' (trans., ''The Clerk'') — “(a special prize commemorating the 400th anniversary of the foundation of Coláiste na nGael in Louvain, awarded under the auspices of the Franciscan Province of Ireland). The prize of €10,000 was the largest prize ever awarded to an Irish language novel [(duais speisialta chomórtha 400 bliain bhunú Choláiste na nGael i Lobháin a bronnadh faoi urraíocht Phroibhinse Phroinsiasach na hÉireann). Ba é an duais €10,000 sin an duais ba mhó a bronnadh riamh ar úrscéal Gaeilge].”&lt;ref name=&quot;:18&quot; /&gt;
* 2021, for ''Súil an Daill'' (''The Eye of the Blind'')
* 2024, for ''Bódléar''

=== Ó Shúilleabháin Award, Irish language “Book of the Year” ===
The first prize of this award includes €5,000 to the publisher and €2,500 to the author of the winning work.&lt;ref name=&quot;:10&quot;&gt;{{Cite journal|date=15 August 2023|title=20 saothar san iomaíocht do ‘Leabhair Ghaeilge na Bliana 2023’|url=https://tuairisc.ie/20-saothar-san-iomaiocht-do-leabhair-ghaeilge-na-bliana-2023/|journal=Tuairisc}}&lt;/ref&gt;

* ''An Cléireach'' (''The Clerk'').&lt;ref&gt;{{Cite web|url=http:/www.gaelport.com/uploads/documents/edition19.html|title=Eagrán / Edition 19 - 04 11 2008|date=4/11/2008|website=Internet Archive|archive-url=https://web.archive.org/web/20130525011340/http:/www.gaelport.com/uploads/documents/edition19.html|archive-date=25 May 2013|access-date=25 August 2025}}&lt;/ref&gt;
* ''Táin Bó Cuailnge'', 2018.
* ''Bódléar'', 2025.

==== De Bhaldraithe Award ====
The Gradam de Bhaldraithe is awarded to the best work in translation.&lt;ref name=&quot;:10&quot; /&gt;

* ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or &quot;'Twas the Night Before Christmas.&quot;&lt;ref name=&quot;:10&quot; /&gt;

==== Other ====

* Walter Macken Prize, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005
* Bháiteir Uí Mhaicín Memorial Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/gaeilge/tuarascail/duais-oireachtais-1.501571|title=Oireachtas Prize: Over €50,000 was awarded to writers in the Oireachtas Literary Competitions at an event in Dublin last night. Winners… [Duais Oireachtais: Bronnadh breis agus €50,000 ar scríbhneoirí i gComórtais Liteartha an Oireachtais ar ócáid i mBaile Átha Cliath aréir. Bhuaigh…]|work=5 October 2005|access-date=15 October 2025}}&lt;/ref&gt;
* BBC Stewart Parker Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2006
* The Aodán Mac Póilín Commemorative Prize, for ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), 2017

== External Links ==

* Leabhar Breac website: https://leabharbreac.com/en/
* Leabhar Breac Facebook pages:
* Rosg website: [http://www.rosg.ie/en/ &lt;nowiki&gt;http://ww&lt;/nowiki&gt;w.rosg.ie/en/]
* Art on the Island (Ealaín ar Oileán) website, archived at the Wayback Machine: https://web.archive.org/web/20130601000520/http://ealainaroilean.ie/ 31 March 2012, 1 June 2013 and 8 January 2014

* Darach Ó Scolaí's website Archived 25 September 2015 at the Wayback Machine: https://web.archive.org/web/20150925103456/http://darachoscolai.ie/
* Youtube video of [https://www.youtube.com/watch?v=OlP2AmSBzXc Breandán Ó Cróinin introducing Deirdre at the book launch] in the pub Tigh Mholly (Molly’s House).

== Primordial Ooze ==

* Known for his sensitivity to language and voices.
* Finish scanning through JSTOR
* Scan through Irish Times, 56 hits
* Check Goodreads
* Check YouTube (In the spring of 2013, the arts programme Imeall interviewed the author on TG4.)
* Check both Wikipedias for pages on the origins of the retold tales (like Deirdre) and link to this article
* Propose link from University of Galway page once Darach’s is up
* Write Irish National Biography (&lt;nowiki&gt;https://www.dib.ie&lt;/nowiki&gt;) to propose an article about Darach once the Wikip article is done? See what they say.
* Link to Ó Scolaí from the Wikipedia
* Make sure links '''to''' Wikipedia in the actual encyclopedia work right

=== Not Placed Yet ===

* &quot;So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]&quot; — &quot;32. An Cléireach – Darach Ó Scolaí (2)&quot; [18 books got 2 votes, and then they're alphabetized by author's last name, so the 32 of 34 doesn't signify the specificity it seems to]&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Ó Murchú|first=Eoin P.|date=9 June 2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil. [The 30 best Irish books for Irish people]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}&lt;/ref&gt;
* &quot;Below is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers.Here is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers [Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí.Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí].&quot; &quot;Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)&quot;&lt;ref name=&quot;:14&quot; /&gt;
* &quot;Ceann eile de bhuaicphointí na hÉigse a bheidh sa seisiún le Darach Ó Scolaí, duine d’úrscéalaithe móra na Gaeilge, agus duine de chomhbhunaitheoirí teach foilsitheoireachta Leabhar Breac. [Another highlight of the Éigse will be the session with Darach Ó Scolaí, one of the great Irish language novelists, and one of the co-founders of the publishing house Leabhar Breac.]&quot;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Nós|date=4 May 2023|title=Éigse na Bruiséile le filleadh i mí na Bealtaine. [Éigse na Bruséile to return in May]|url=https://nos.ie/cultur/eigse-na-bruiseile-le-filleadh-i-mi-na-bealtaine/|journal=Nós}}&lt;/ref&gt;

=== Things Taken Out for Now ===
“’The play is a comedy about language, lies, bureaucracy and Gaeltacht grants, in the tradition of Myles na Gcopaleen,’ according to Norma-Jean Kenny in the ''Galway Advertizer'', ‘in which the author comments and criticizes the institutions of the Irish language in Ireland without ceasing.’&quot;

Supposedly a quotation by Gearóid Denvir reviewing ''Táin Bó Cuailnge'' for ''Comhar'' (but I don't find it in the article):

This book has long been needed by Irish language readers and there is no doubt that it will become a classic in time and surpass Thomas Kinsella’s English version. This version remains faithful to the language of the original while at the same time finding an appropriate language in today’s Irish. Ó Scolaí masterfully overcomes the difficulties of the original’s rhetorical difficulties and the versions of the original poetic texts are extremely effective.[supposedly &lt;ref name=&quot;:15&quot; /&gt;]

“The biggest prize ever awarded for a novel in Irish was presented at a special ceremony in the National Concert Hall in Dublin, today (Thursday, 4 October 2007). Darach Ó Scolaí, writer, artist &amp; playwright from Casla, Co. Galway, was awarded €10,000 for his literary novel, ‘An Ardscoil’. This work, under the new title ‘An Cléireach’, will be launched at Oireachtas na Samhna in Westport in November. This is the first novel from his pen, a story set in the late seventeenth century. This competition was sponsored by the Franciscan Province of Ireland.” (archive, Oireachtas na Gaeilge site, 04 October, 2007)

''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. number 2 in ''Comhar'' literary magazine’s list of best books of 2021. '''{6}.'''

*William Shakespeare, ''Romeo agus Juliet'' (trans. of ''Romeo and Juliet''), Leabhar Breac, 2016.
*Jonathan Swift, ''Camchuairt Ghuilivéir'' (trans. of ''Gulliver's Travels''), Leabhar Breac, 2016.

*Hugo Pratt, ''Corto Maltese''


'''''Flag of Bones (Bratach na gCnámh) Series'''''

Leabhar Breac published the Bratach na gCnámh series of books for young readers. Written in French by Alain Surget, illustrated by Annette Marnat and translated by Darach Ó Scolaí, this series uses the history of Caribbean Sea pirates&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/product-category/alain-surget/|title=Alain Surget Archives|website=Leabhar Breac|language=en-US|access-date=2025-09-30}}&lt;/ref&gt;:

*Alain Surget, ''Éalú as Páras'' (''Escape from Paris''), Annette Marnat (Illustr.), Leabhar Breac, 2011.
* Alain Surget, ''Oilean na Siorcanna'' (''Shark Island''), Annette Marnat (Illustr.), Leabhar Breac, 2011.
* Alain Surget, ''Long na dTaibhsi'' (''Ship of the Ghosts''), Annette Marnat (Illustr.), Leabhar Breac, 2011.
* Alain Surget, ''San Ochtapas Dubh'' (''In the Black Octopus''), Annette Marnat (Illustr.), Leabhar Breac, 2013.
* Alain Surget, ''San Ionsai ar Veracruz'' (''The Attack on Veracruz''), Annette Marnat (Illustr.), Leabhar Breac, 2013.
'''''For &quot;First Readers&quot; (children to 6 years old or so)'''''

These books were written originally in Catalan by Spanish author Enric Lluch Girbés and translated into Irish by Ó ScolaÍ:
*Enric Lluch, ''Ag Péinteáil an Tí'' (''Painting the House''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''An Colúr Bacach'' (''The Lazy Dove''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''An Phluais'' (''The Cave''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''Madra Dhaideo'' (''Grandpa's Dog''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''Fiacail Mháire'' (''Mary's Tooth''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
'''''Bruno Heitz'''''

Leabhar Breac published a series of 3 Heitz books for small children. Published originally in French, this series of three comic books is about a blind mole named Cáitín Chaoch in Irish (and ''Louisette la taupe'' in French).&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/product-category/bruno-heitz-en/|title=Bruno Heitz Archives|website=Leabhar Breac|language=en-US|access-date=2025-10-02}}&lt;/ref&gt; Ó Scolaí translated these:

*Bruno Heitz (author and illustr.), ''Práinneach'' (''Urgent''), Leabhar Breac, 2020
*Bruno Heitz (author and illustr.), ''Preab san Aer'' (''Bounce in the Air''), Leabhar Breac, 2020.

'''''Books for Toddlers'''''

Leabhar Breac has published 14 books written by French author Orianne Lallemand's and illustrated by Eleonore Thuillier, about Lallemmand's popular character Loup, Wolf. These are translated by Ó Scolaí:
*Orianne Lallemand, ''An Mac Tire a Raibh Faitios an Domhain Air'' (trans. of ''The Son Who Saw the World in His Eyes''), Eleonore Thuillier  Illustr.), Leabhar Breac, 2018.
*Orianne Lallemand, ''Macan agus an Goban'' (trans. of ''Macan and the Goblin''), Eleonore Thuillier (Illustr.), Leabhar Breac, 2018.
* Orianne Lallemand, ''A Mac Tíre a Chuaigh go Tóin na Farraige'' (trans. of ''The Wolf Who Went to the Bottom of the Sea''), Éléanore Thuillier  (Illustr.), Leabhar Breac, 2019.
'''''Board Books (for babies)'''''

J. C. (Joan Carles) Girbés Aparisi is a Catalan author and editor. These books were written in Catalan and translated by Ó Scolai.
*J. C. Girbés, ''An Phicnic'' (''The Picnic''), Silvia Ortega (Illustr.), Leabhar Breac, 2013.
* J. C. Girbés, ''An Chóisir'' (''The Party''), Silvia Ortega (Illustr.), Leabhar Breac, 2013.
*J. C. Girbés, ''Lá Mór Fada'' (''A Long Day''), Silvia Ortega (Illustr.), Leabhar Breac, 2014.
*J. C. Girbés, ''Tabhair Leat do Leabhar'' (''Bring Your Book''), Silvia Ortega (Illustr.), Leabhar Breac, 2014.

==== Gradam Réics Carló ====
The Réics Carló prize is awarded for the best book in the Irish language for young readers. It is named for one of the characters of 20th-century writer [[wikipedia:Cathal_Ó_Sándair|Cathal Ó Sándair (Charles Saunders)]].

* ''An Bradán Feasa'' was “shortlisted for the Réics Carlo award 2010.”&lt;ref name=&quot;:9&quot; /&gt;

== References ==
{{reflist}}</text>
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{{Motivation and emotion/Tutorials/Complete2}}
&lt;!-- {{Motivation and emotion/Tutorials/Complete2}} --&gt;
&lt;!-- {{Motivation and emotion/Tutorials/In development}} --&gt;

==Overview==
This tutorial facilitates an interactive discussion about two aspects of [[self-determination theory]]:
# basic psychological needs
# taxonomy of motivation (intrinsic-extrinsic motivation)

==Needs==
Brainstorm and discuss:

{{collapse box|'''What are needs?'''|Needs are ''requirements'' (essential) for surviving (biological needs) and thriving (psychological needs).}}
{{collapse box|'''How do needs differ from desires, wants, likes etc.?'''|Needs are ''essential'' for survival (e.g., warmth, transport) and well-being whereas desires, wants, likes etc. are motivations for ''non-essential'' stimuli (e.g., an umbrella in the rain, a new car).}}
{{collapse box|'''What types of needs are there?'''|'''2-factor''': deficit-based (biological) needs (e.g., thirst, hunger, sleep) and growth-based (psychological) needs (e.g., autonomy, competence, relatedness).&lt;br&gt;'''3-factor''': The Alderfer ERG theory refers to existence, relatedness, and growth needs.&lt;br&gt;'''5-factor''': Maslow's hierarchy of needs has five needs: Physiological, Safety, Belonging &amp; love, Esteem, Self-actualisation}}
{{collapse box|'''What are the basic psychological needs?'''|According to self-determination theory:&lt;br&gt;autonomy, competence, and relatedness}}

==Taxonomy of motivation==
Discuss:

{{collapse box|'''What characterises [[extrinsic motivation]]? Examples?'''|
* External (environmental) reasons for behaving
* Caused by carrots (incentives and rewards e.g., praise) or sticks (threats and punishments e.g., insults).}}
{{collapse box|'''What characterises [[intrinsic motivation]]? Examples?'''|
* Internal reasons for behaving
* Involves volition (i.e., ''I want to'') and enjoyment (e.g., hobbies)
* External influences are absent.}}
{{collapse box|'''Advantages of the I-E motivation distinction?'''|
* Breaks motivation down into types
* Simple and easy to understand
* Helps with research designs}}
{{collapse box|'''Disadvantages of the I-E motivation distinction?'''|
* A dichotomous distinction is overly simplistic
* Motivational sources are often complex and dynamic, involving aspects of both intrinsic and extrinsic motivation}}
{{collapse box|'''Types of motivation described by [http://mmrg.pbworks.com/f/Ryan,+Deci+00.pdf Deci and Ryan (2000)]?'''|
* Amotivation
* Extrinsic motivation
** External regulation
** Introjection
** Identification
** Integration
* Intrinsic motivation
}}
&lt;!-- Handout a [https://drive.google.com/file/d/0B2N4zSp4hmN9WHEwSmoyUWhOZm8/view?usp=sharing blanked version of Deci and Ryan's taxonomy] and facilitate the class to define, discuss, and generate examples for each of the components:--&gt;

== Quiz: Identify the type of motivation ==
This 12-item quiz tests your knowledge of Deci and Ryan's taxonomy of motivation.

[[/Quiz|Go to quiz]].

Alternative (advenced) exercise: What strategies can motivate exercise behaviours?
* Brainstorm ways of motivating healthy exercise behaviours.
* Allocate each of these strategies along the taxonomy of motivation, including amotivation, the four types of extrinsic motivation, and intrinsic motivation.

'''Table 1'''.

What Might Someone With Each of These Six Types of Motivation Say about Their Reasons for Exercising?

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
! Type !! Example reason
|-
| '''Amotivation'''
| &quot; ......... ? ......... &quot;
|-
| '''External regulation'''
| &quot; ......... ? ......... &quot;
|-
| '''Introjection'''
| &quot; ......... ? ......... &quot;
|-
| '''Identification'''
| &quot; ......... ? ......... &quot;
|-
| '''Integration'''
| &quot; ......... ? ......... &quot;
|-
| '''Intrinsic motivation'''
| &quot; ......... ? ......... &quot;
|}

For examples, see Table 2.

==Answers==
'''Table 2'''.

Examples of Reasons to Engage in Healthy Exercise Based on the Taxonomy of Motivation (Deci &amp; Ryan, 2000)

{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
! Type !! Example reason
|-
| '''Amotivation'''
| &quot;I don't see any point in exercising because it doesn't make a difference.&quot;
|-
| '''External regulation'''
| &quot;My doctor told me I have to exercise.&quot;
|-
| '''Introjection'''
| &quot;I feel guilty when I don't exercise.&quot;
|-
| '''Identification'''
| &quot;Being fit is important to me.&quot;
|-
| '''Integration'''
| &quot;Taking care of my body is part of who I am.&quot;
|-
| '''Intrinsic motivation'''
| &quot;I exercise because I genuinely enjoy being active.&quot;
|}

==Book chapter ==
===Topic development feedback===
* Review [[Motivation and emotion/Assessment/Topic|topic development]] marks and feedback – follow-up and discuss if you don’t understand
* '''Align''' sub-title, focus questions, and headings
* Identify the '''best psychological science on the topic''': Systematic reviews and/or meta-analysis
* '''Embed''' links to more information:
** Other book chapters
** Wikipedia articles
* '''Acknowledge use of genAI''' in '''edit summaries''' by including a '''shared link''' to the conversation (be academically honest and transparent). See [[Motivation and emotion/Assessment/Using generative AI|using generative AI guidelines]].

* [[Motivation and emotion/Assessment/Topic/Feedback|Topic development - General feedback]]

===Messy middle===
We are now in the &quot;messy middle&quot; of the [[Motivation and emotion/Assessment/Chapter|book chapter]] development:
* '''Motivational pattern''' (consistent with motivational science about the trajectory of motivation)
** Strong at '''start''' (excitement, newness)
** Strong at '''end''' (deadline pressure, completion goals)
** Dips in '''middle''' (novelty fades, finish feels long, distractions increase)
* '''Strategies for the middle'''
** Make '''small, regular edits''' → steady progress
** Use '''scaffolding''' → build on [[Motivation and emotion/Assessment/Topic|topic development]] feedback and work towwards [[Motivation and emotion/Assessment/Chapter|chapter guidelines]]
** Seek clarification; follow up if unsure
* '''Key idea'''
** Keep '''momentum''' alive in the “messy middle”—in this case, by doing small edits regularly to move from Topic Development → Book Chapter

==Recording==
* [https://au-lti.bbcollab.com/recording/56e88f2f9a5a4977972c3adcf0b116fb Tutorial 4] (2026)&lt;!--
* [https://au-lti.bbcollab.com/recording/fecbd3c2b9c94ee8bf7aaf6730226a53 Tutorial 4] (2025)
* [https://au-lti.bbcollab.com/recording/9717613592cf458099da1b8e021fe13f Tutorial 4] (2024)
* [https://au-lti.bbcollab.com/recording/5a95256f932d4e228992199adcb6c094 Tutorial 4] (2023)
* [https://au-lti.bbcollab.com/recording/8eeaa7d6c18c4efbafc2c28984bfbaca Tutorial 4] (2022)
* [https://au-lti.bbcollab.com/recording/ee193c24a6414276b3d47acba500d433 Tutorial 4] (2021) --&gt;

==See also==
* [[/Instructor notes/]]
;Additional tutorial material
* [[/Example of self-determination theory: Child cleaning teeth/|Example of SDT: Child cleaning teeth]]
* [[../Needs/Implicit motives/|Implicit motives]]
* [[../Needs/Models which provide organisation of needs/|Models which organise needs]]
* [[/Self-determination theory/]]
&lt;!--
;Book chapters
--&gt;
&lt;!--
;Wikipedia
--&gt;
; Lecture
* [[Motivation and emotion/Lectures/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]]

;Tutorials
* [[{{#titleparts:{{PAGENAME}}|2}}/Physiological needs|Physiological needs]] (Previous tutorial)
* [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking|Functionalist theory and self-tracking]] (Next tutorial)
&lt;!--
;Admin
* [[/Instructor notes/]]
--&gt;

==References==
{{Hanging indent|1=Deci, R. M., &amp; Ryan, E. L. (2000). Intrinsic and extrinsic motivations: Classic definitions and new directions. ''Contemporary Educational Psychology'', ''25''(1), 54–67. https://doi.org/10.1006/ceps.1999.1020 [https://www.sciencedirect.com/science/article/pii/S0361476X99910202/pdf?md5=b9e2589a068cf7f9529c1bff165ddd55&amp;pid=1-s2.0-S0361476X99910202-main.pdf PDF]}}

==External links==
* [http://www.youtube.com/watch?v=VGrcets0E6I Promoting motivation, health, and excellence] (Ed Deci, 2012, TEDx, YouTube 14:05 mins)
* [https://selfdeterminationtheory.org/ Self-determination theory] (selfdeterminationtheory.org)
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See also [[Universal Bibliography/Geography|Geography]].

See [[w:Category:Bibliographies of countries or regions]] and [[w:Category:Works about countries]].

This part of the [[Universal Bibliography]] is a bibliography of countries (including former countries).

==Countries==
*Bateman and Egan (eds). The Encyclopedia of World Geography: A Country by Country Guide. 1993. Revised 1997.
*Peter Stalker. Handbook of the World. 2000. A Guide to Countries of the World. (Oxford Guide to Countries of the World. 2nd Ed: 2004, 2nd Revised Ed: 2007 [https://books.google.co.uk/books?id=GtztAAAAMAAJ], 3rd Ed: 2010 [https://books.google.co.uk/books?id=gvKvfxkbZ1AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Countries of the World and Their Leaders Yearbook. Gale. [https://books.google.co.uk/books?id=5etKAAAAYAAJ] [https://books.google.co.uk/books?id=p41OAAAAIAAJ]
*Hutchinson Guide to Countries of the World [https://books.google.co.uk/books?id=GgpjUe4kN_IC]
*The World Guide: Global Reference, Country by Country. 11th Ed: 2007 [https://books.google.co.uk/books?id=EoWoLgAACAAJ]
*Spence. The World Today: A Nation-by-Nation Guide. Cassell. 1994. 1999. [https://books.google.com/books?id=Ub8qOQAACAAJ]
*Worldmark Encyclopedia of the Nations [https://books.google.co.uk/books?id=I0oYAQAAMAAJ]
*Kurian. Encyclopedia of the World's Nations. Facts on File. Reviews: [https://books.google.co.uk/books?id=Y1EnAQAAIAAJ] [https://books.google.co.uk/books?id=lz0RAQAAMAAJ]
*Michael O'Mara. Facts about the World's Nations. 1999. [https://books.google.co.uk/books?id=mygYAAAAIAAJ]
*Status of the World's Nations. 1965 [https://books.google.co.uk/books?id=sftEyRbAXMUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 1973 [https://books.google.co.uk/books?id=kw2U_Cg2gKYC&amp;pg=PP3#v=onepage&amp;q&amp;f=false].
*[[s:Author:John Alexander Hammerton|Hammerton, John Alexander]] (ed). Countries of the World. Published at the Fleetway House. 6 vols. [https://books.google.co.uk/books?id=e6IaAQAAMAAJ] [https://books.google.co.uk/books?id=K5oaAQAAMAAJ]
*[[s:Author:Robert Brown (1842-1895)|Brown, Robert]]. The Countries of the World. [https://books.google.co.uk/books?id=nO0DAAAAQAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*A Morely Dell. The Countries of the World. (Harrap's New Geographical Series). 1932. (School certificate). Reviews: [https://books.google.co.uk/books?id=oSS9PB_Jf7AC] [https://books.google.co.uk/books?id=BicVAAAAIAAJ] [https://books.google.co.uk/books?id=5qBOAAAAIAAJ] [https://books.google.co.uk/books?id=YbwcAQAAIAAJ] [https://books.google.co.uk/books?id=sc1AAAAAIAAJ]

General series:
*National Geographic Countries of the World [https://books.google.co.uk/books?id=IT2wfzVIPykC]
*Countries of the World. Evans Brothers. (GCSE) [https://books.google.co.uk/books?id=a3sZvWc7E1EC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*One Europe. Longman. [https://search.worldcat.org/en/title/west-germany-adapted-by-lj-russon-from-the-original-german-by-sylvia-lof-ingrid-mallberg-dietrich-rosenthal/oclc/561591761]
*Collier's Nations of the World. The Nations of the World: An Historical Series. [https://books.google.co.uk/books?id=VJY-AAAAYAAJ&amp;pg=PP8#v=onepage&amp;q&amp;f=false]
*Collier's History of Nations. The History of Nations. [https://books.google.co.uk/books?id=fmSUfTY5E80C]
*The Story of the Nations. T Fisher Unwin.
*The World and Its Peoples. (The Illustrated Library of the World and Its Peoples). Greystone Press, New York.
*World and Its Peoples. Marshall Cavendish. [https://books.google.co.uk/books?id=oms5xjI7ba0C&amp;pg=PA141#v=onepage&amp;q&amp;f=false]

==England==
===Counties===
See [[s:Portal:Counties]]

* Harrison, &quot;County Bibliography&quot; (1886) 3 Library Chronicle [https://books.google.co.uk/books?id=Wz9FAAAAYAAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false 49]

General series
*Victoria County History
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

Avon
*Moore. Avon Local History Handbook. Phillimore. 1979. [https://books.google.co.uk/books?id=h0kjAAAAMAAJ] Bibliography, p 102

Bedfordshire
*Conisbee, Lewis Ralph. A Bedfordshire Bibliography. Bedfordshire Historical Record Society. Bedford. 1962. Supplements 1967, 1971, 1978. Third supplement by Threadgill. Review: 6 Archives 52 [https://books.google.co.uk/books?id=oOMZAAAAYAAJ]. See also [https://books.google.co.uk/books?id=MjspAAAAYAAJ] [https://books.google.co.uk/books?id=PejgAAAAMAAJ]
*Godber. History of Bedfordshire. 1969. 1984. [https://books.google.co.uk/books?id=jdvwPQAACAAJ]
*Pinnock. The History and Topography of Bedfordshire [https://books.google.co.uk/books?id=9bJYAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Parry. Select Illustrations, Historical and Topographical, of Bedfordshire [https://books.google.co.uk/books?id=UTUJAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Blyth. The History of Bedford and Visitor's Guide. 1873 [https://books.google.co.uk/books?id=IuIGAAAAQAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Cambridge County Geographies [https://books.google.co.uk/books?id=kTc8AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Buckinghamshire
*Reed. A History of Buckinghamshire. 1993 [https://books.google.co.uk/books?id=BtkWAQAAIAAJ]

Cambridgeshire
*Carter. History of the County of Cambridge [https://books.google.co.uk/books?id=jXpbAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Babington. Ancient Cambridgeshire [https://books.google.co.uk/books?id=DPrCAwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Devon
*Ravenhill and Rowe. Devon Maps and Map-makers [https://books.google.co.uk/books?id=tjf2yAEACAAJ]
*Wright. A Plea for a Devonshire Bibliography. 1885 [https://books.google.co.uk/books?id=8ZUDAAAAQAAJ]

Derbyshire
*Woore. A Catalogue of Local Maps of Derbyshire, C.1528-1800. 2012. [https://books.google.co.uk/books?id=oWmCMwEACAAJ]
*O'Neal. A Bibliography of Derbyshire Lead Mining. 1961

Essex
*Cunnington. Catalogue of Books, Maps and Manuscripts, relating to or connected with the County of Essex. 1902 [https://books.google.co.uk/books?id=oIcqpibGE4MC]
*&quot;The Bibliography of Essex&quot; (1882) 1 Antiquarian Magazine &amp; Bibliographer [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false 72]. See also [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA283#v=onepage&amp;q&amp;f=false p 283].
*&quot;The Bibliography of Essex&quot; (1891) 5 The Essex Naturalist 30 [https://books.google.co.uk/books?id=iIo1AQAAMAAJ]
*Moon. Essex Literature. 1900. Review: 61 Literary World 438 [https://books.google.co.uk/books?id=2T0ZAAAAYAAJ] See also [https://books.google.co.uk/books?id=1Y4UAQAAIAAJ] [https://books.google.co.uk/books?id=C_pEAAAAMAAJ]
*Fenn and Lowery, &quot;An Essex Bibliography&quot;, Journal of the South West Essex Technical College, vols 2 &amp; 3
*Victoria County History bibliography. 1959 [https://books.google.co.uk/books?id=F2EJAQAAIAAJ]
*O'Leary, John Gerard. A Supplement to the Essex Bibliography. Dagenham. 1962.
*A Bibliography of Essex Archaeology &amp; History
*Essex and Dagenham: A Catalogue of Books, Pamphlets and Maps. Dagenham. 1961
*Essex Archaeology and History: The Transactions of the Essex Society for Archaeological and History [https://books.google.co.uk/books?id=CtFAAAAAYAAJ]
*Essex Naturalist: Being the Journal of the Essex Field Club
*Wright. The History and Topography of the County of Essex [https://books.google.co.uk/books?id=SgQVAAAAQAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Ogborne, The History of Essex [https://books.google.co.uk/books?id=IeVSAAAAcAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Suckling. Memorials of the Antiquities and Architecture, Family History and Heraldry of the County of Essex [https://books.google.co.uk/books?id=bcw_AAAAcAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Hunter, The Essex Landscape: A Study of Its Form and History [https://books.google.co.uk/books?id=w9kWAQAAIAAJ]
*Cambridge County Geography [https://books.google.co.uk/books?id=GPHa_X_0qo0C&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Sokoll. Essex  Pauper Letters, 1731-1837 [https://books.google.co.uk/books?id=rCLia7XlqtMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Morant. The History and Antiquities of Colchester in the County of Essex [https://books.google.co.uk/books?id=DDgtAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Wallen. The History and Antiquities of the Round Church at Little Maplestead, Essex [https://books.google.co.uk/books?id=FPYVAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Kent
*Smith. Bibliotheca Cantiana. 1837. [https://books.google.co.uk/books?id=1dJDAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

Leicestershire
*Kirkby, C V (compiler). Catalogue of the books, pamphlets, &amp;c., relating to Leicestershire in the Central Reference Library. Leicester Free Public Libraries. 1893. Reviews: [https://books.google.co.uk/books?id=3boqAQAAIAAJ&amp;pg=PA84#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=UcHnAAAAMAAJ&amp;pg=PA728#v=onepage&amp;q&amp;f=false]
*Leicestershire and Rutland Bibliography, 1963-65 (1966) [https://books.google.co.uk/books?id=-OhVAAAAYAAJ 40] Leicestershire Archaeological and Historical Society: Transactions (1964/5) 92. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1961-63. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1960-61. Available as pdf from University of Leicester.
*A Bibliography of the Small Towns in Leicestershire and Rutland, 1600–1850. (Dissertation). [https://repository.lboro.ac.uk/articles/educational_resource/A_bibliography_of_the_small_towns_in_Leicestershire_and_Rutland_1600_1850/9414200]
*Loughborough's Heritage: A Bibliography of the Holdings of Leicestershire Libraries and Information Service and Record Office. [https://books.google.co.uk/books?id=Bwx2zgEACAAJ]
*Keith Ambrose and Frank Williams, &quot;Bibliography of the Geology of Leicestershire and Rutland: Part 2: 1971-2003&quot; (2004) [https://books.google.co.uk/books?id=U-tQAQAAIAAJ 16] The Mercian Geologist 5. Available as pdf from East Midlands Geological Society.
*Parsons and Brandwood. A Bibliography of Leicestershire Churches. 1978.
*Education in Leicestershire: A Bibliography. [https://books.google.co.uk/books?id=X6EfzQEACAAJ]

Sussex
*Brent, Fletcher and McCann. Sussex in the 16th and 17th Centuries: A Bibliography. 2nd Ed [https://books.google.co.uk/books?id=I7UtAAAAYAAJ]
*Farrant. Sussex in the 18th and 19th Centuries: A Bibliography. 1st Ed: 1973, 2nd Ed: 1977 [https://books.google.co.uk/books?id=MLUtAAAAYAAJ], 3rd Ed: 1979

==France==
Bibliography:
*Bibliographie de la France. Commentary: Encyclopedia of Library and Information Science, vol 37, supplement 2, [https://books.google.co.uk/books?id=10rgjNvOV8oC&amp;pg=PA145#v=onepage&amp;q&amp;f=false p 145]; The Bookseller, 6 January 1881, [https://books.google.co.uk/books?id=4dsiAQAAMAAJ&amp;pg=PA10#v=onepage&amp;q&amp;f=false p 10]; Stein, Manuel de bibliographie générale, [https://books.google.co.uk/books?id=lJYPyKjV1qYC&amp;pg=PA23#v=onepage&amp;q&amp;f=false p 23]. 
*Girault de Saint-Fargeau. Bibliographie historique et topographique de la France. 1845 [https://books.google.co.uk/books?id=kClB9CQNZoMC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Catalogue d'une collection d'ouvrages sur l'histoire des provinces de la France. 1842 [https://books.google.co.uk/books?id=qQBX5WZouzAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Landscape:
*Beaujeu-Garnier. France. (The World's Landscapes). 1975. [https://books.google.com/books?id=nwxDAQAAIAAJ]

Agenais:
*Andrieu. Bibliographie générale de l’Agenais et des parties du Condomois et du Bazadais. 1886 to 1891. Reprinted 1969.

Alsace:
*Ristelhuber. Bibliographie alsacienne. 1869 to 1873 [https://books.google.co.uk/books?id=0mhLAQAAMAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Bibliographie alsacienne: Revue critique des publications concernant l'Alsace. 1918 to 1936
*Ritter. Répertoire bibliographique des livres imprimés en Alsace aux XVe et XVIe siècles [https://books.google.co.uk/books?id=DewaAQAAMAAJ]

Angoumois:
*Castaigne. Essai d'une bibliothèque historique de l'Angoumois, ou Catalogue raisonné des principaux ouvrages qui traitent des différentes branches de l'histoire de cette province. 1847 [https://books.google.co.uk/books?id=R-UanmmlvAEC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Anjou:
*Braguier and Braguier. Archéologie en Anjou: bibliographie. 1984 [https://books.google.co.uk/books?id=LvsmAQAAIAAJ]

Auvergne:
*Gonot. Catalogue des ouvrages imprimés et manuscrits concernant l'Auvergne, extrait du catalogue général de la Bibliotlèque de Clermont-Fd (Puy-de-Dome). 1849. [https://books.google.co.uk/books?id=yCFtbObRCbUC&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Catalogue des livres et estampes concernant l'ancienne Province d'Auvergne (Puy-de-Dôme, Cantal, Haute-Loire) réunis par feu M. G. Desbouis. 1865. [https://books.google.co.uk/books?id=Ui4S8_D0N74C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Béarn
*&quot;Bibliographie Béarnaise&quot;, Revue de Pau et du Béarn [https://books.google.co.uk/books?id=FuZnAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=FQYqvPo9D9IC&amp;pg=PA158#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RL9VAAAAYAAJ]

Brittany
*Sacher. Bibliographie de la Bretagne, ou Catalogue général des ouvrages historiques, littéraires et scientifiques parus sur la Bretagne, avec la liste des revues publiées en cette province, les prix approximatifs des volumes rares, etc. 1881 [https://archive.org/details/bibliographiede00sach]

Burgundy:
*Milsand. Bibliographie bourguignonne; ou, Catalogue méthodique d'ouvrages relatifs à la Bourgogne: Sciences - Arts - Histoire. 1885 [https://archive.org/details/bibliographiebo00milsgoog] [https://archive.org/details/bibliographiebo00sciegoog] [https://books.google.co.uk/books?id=CxIIAAAAQAAJ]
*Catalogue des manuscrits de la Bibliothèque royale des ducs de Bourgogne. 1842 [https://books.google.co.uk/books?id=FX5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*The Companion Guide to Burgundy [https://books.google.co.uk/books?id=NraRP0AkDT0C&amp;pg=PP3#v=onepage&amp;q&amp;f=false]
*Lecat. The Golden Book of Burgundy. (The Golden Book) [https://books.google.co.uk/books?id=FyzR9qU1Zl4C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Gwynn. Burgundy: With Chapters on the Jura and Savoy. (Kitbag Travel Books). 1935 [https://books.google.co.uk/books?id=ny1LAAAAMAAJ]
*Bazin. Wonderful Burgundy. 1988. 1997 [https://books.google.co.uk/books?id=Yt1CRdICWCUC]
*Bailey. Burgundy. (Insight Guides). 1993 [https://books.google.co.uk/books?id=Q69a1dMW2NQC]
*Dunlop. Burgundy. Hamilton.1990 [https://books.google.co.uk/books?id=S_1OAAAAMAAJ]

Champagne:
*Lhermitte. Ouvrages sur la Champagne: contribution à la bibliographie champenoise. 1992. [https://books.google.co.uk/books?id=jbPfAAAAMAAJ]

Dauphiné:
*Mélanges biographiques et bibliographiques relatifs à l'histoire littéraire du Dauphiné par Colomb de Batines et Ollivier Jules. 1837 [https://books.google.co.uk/books?id=2F5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Lorraine:
*Bibliographie lorraine. Académie nationale de Metz [https://books.google.co.uk/books?id=n-DfAAAAMAAJ]

Maine:
*Desportes. Bibliographie du Maine, précédée de la description topographique et hydrographique du diocése du Mans, Sarthe et Mayenne. 1844. [https://books.google.co.uk/books?id=hSk-AAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Normandy:
*Frère. Manuel du bibliographe Normand ou dictionnaire bibliographique et historique. 1858 to 1860. [https://books.google.co.uk/books?id=dp6geJClg1YC&amp;pg=PP13#v=onepage&amp;q&amp;f=false vol 1]

==Japan==
Bibliography and literature
*Hideo Kaneko. &quot;Japanese Literature and Bibliography&quot;. Kent, Lancour and Daily (eds). Encyclopedia of Library and Information Science. Marcel Dekker. 1977. vol 21. pp [https://books.google.co.uk/books?id=H1pNvzr_n98C&amp;pg=PA131#v=onepage&amp;q&amp;f=false 131] to 176.

Bibliography
*Jozef Rogala. A Collector's Guide to Books on Japan in English: An Annotated List of Over 2500 Titles with Subject Index. 2001. [https://books.google.co.uk/books?id=7KI9ao-w2FEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ria Koopmans-de Bruijn. Area Bibliography of Japan. (Scarecrow Area Bibliographies). Scarecrow Press. 1998. [https://books.google.co.uk/books?id=Hlx2OMjgUi0C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Frank Joseph Shulman. Japan. (World Bibliographical Series, vol 103). Clio Press. 1989. [https://books.google.co.uk/books?id=LsoUAQAAIAAJ]
*Eibun Nihon Kankei Tosho Mokuroku, 1945-1981. (Japanese: 英文日本関係図書目録, 1945-1981). (English: Catalogue of Books in English on Japan, 1945-1981). Japan Foundation. Tokyo. 1986.
*Japan: analytical bibliography: with supplementary research aids: and selected data on Okinawa . . . Department of the Army. Washington. 1972. [https://books.google.co.uk/books?id=h4d4nYxrxtMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Books on Japan in Western Languages. The International Christian University Library. 1971. [https://books.google.co.uk/books?id=F2bQAAAAMAAJ]
*Books on Japan: A List of Acquisitions, 1955-1970. International House of Japan Library. 1971. [https://books.google.co.uk/books?id=F8sWAQAAIAAJ]
*Fukuda. Union Catalog of Books on Japan in Western Languages. 1968. [https://books.google.co.uk/books?id=HKYyAQAAIAAJ]
*A Classified List of Books in Western Languages Relating to Japan. University of Tokyo Press. 1965. [https://books.google.co.uk/books?id=U8MUAQAAIAAJ]
*Katsuji Yabuki (ed). Japan Bibliographic Annual. Published by the Hokuseido Press for the Japan Writers Society. 1956 and 1957.
**Japan Bibliographic Annual 1956. [https://books.google.co.uk/books?id=9XLQAAAAMAAJ]
**Japan Bibliographic Annual 1957. [https://books.google.co.uk/books?id=vesSAAAAIAAJ]. Reviews: (1957) 13 Monumenta Nipponica 166 (April-July) [https://books.google.co.uk/books?id=8S1yb-iwrOwC] (1957) 25 The Oriental Economist 212 (April) [https://books.google.co.uk/books?id=QELoAAAAMAAJ]
*Haring. Books on Japan: A Reference List. 1955. [https://books.google.co.uk/books?id=RbDoAAAAMAAJ]
*Borton. A Selected List of Books and Articles on Japan in English, French, and German. 1940: [https://books.google.co.uk/books?id=YYIsAAAAYAAJ]. Revised and enlarged. Harvard University Press. 1954: [https://books.google.co.uk/books?id=F8O2VwJUPUkC].
**A Selected List of Books on Japan in Western Languages (1945-1960). (Studies on Asia Abroad, vol 1). The Information Centre of Asian Studies, The Toyo Bunko. 1964. [https://books.google.co.uk/books?id=i1_QAAAAMAAJ]
*Oskar Nachod. Bibliography of the Japanese Empire 1906-1926. 1928. [https://archive.org/details/bibliographyofja0001oska/page/n8/mode/1up vol 1]. [https://archive.org/details/bibliographyofja0002oska/page/n6/mode/1up vol 2].
*Fr. von Wenckstern. A Bibliography of the Japanese Empire: being a Classified List of All Books, Essays and Maps in European Languages relating to Dai Nihon (Great Japan) published in Europe, America and in the East from 1859-93 . . . 1895. vol 1. [https://books.google.co.uk/books?id=dcVAAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=v7lO4ddqDywC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**Volume 2, from 1894 to the middle of 1906. 1907. [https://archive.org/details/bibliographyofja0002frvo/page/n6/mode/1up]
*Hyman Kublin. What Shall I Read on Japan? An Introductory Guide. Japan Society, New York. 1971. [https://books.google.co.uk/books?id=yRRUAAAAYAAJ]

Japanese studies
*An Introductory Bibliography for Japanese Studies. The Japan Foundation. [https://books.google.co.uk/books?id=53O6AAAAIAAJ]
*Richard Perren. Japanese Studies from Pre-History to 1990: A Bibliographical Guide. 1992. [https://books.google.co.uk/books?id=CN9RAQAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies&quot; at pp 1 to 3.
*K.B.S. Bibliography of Standard Reference Books for Japanese Studies, with Descriptive Notes. University of Tokyo Press. [https://books.google.co.uk/books?id=95wbAAAAMAAJ]
*[[w:en:Japan Forum|Japan Forum]]. British Association for Japanese Studies. [https://www.tandfonline.com/journals/rjfo20]

History and culture
*John W Dower. Japanese History &amp; Culture from Ancient to Modern Times: Seven Basic Bibliographies. 1986. [https://books.google.co.uk/books?id=NX67AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies &amp; Research Guides&quot; at chapter 6.

Research guides
*Mindy L Kotler. Information Gathering on Japan: A Primer. Search Associates. 1988. ISBN 9780962546006. Catalogue: [https://search.worldcat.org/zh-cn/title/Information-gathering-on-Japan-Joho-:-a-primer/oclc/20530148]. Review: (1989) [https://books.google.co.uk/books?id=NZLiAAAAMAAJ 27] Choice 82

Encyclopedias

See also [[w:ja:Japanese encyclopedias]]
*Louis-Frédéric. Japan Encyclopedia. 2002. [https://books.google.co.uk/books?id=p2QnPijAEmEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Japan: An Illustrated Encyclopedia. Kodansha. 1993.
**Japan: Profile of a Nation. Kodansha. 1995. Revised Edition. 1999.
*[[w:Kodansha Encyclopedia of Japan|Kodansha Encyclopedia of Japan]]. 1983. Supplement. 1986. [https://books.google.co.uk/books?id=WvApAQAAMAAJ]
*Dorothy Perkins. Encyclopedia of Japan: Japanese History and Culture, from Abacus to Zori. Facts on File. A Roundtable Press Book. 1991. [https://books.google.co.uk/books?id=JLKGAAAAIAAJ]
*Pictorial Encyclopedia of Modern Japan. Gakken. 1986. [https://books.google.co.uk/books?id=0FgKAQAAIAAJ]
*Boye Layfayette De Mente. Japan Encyclopedia. 1995. [https://books.google.co.uk/books?id=f9c7AAAAMAAJ]
**Boye De Mente. Everything Japanese. [The Authoritave Reference on Japan Today]. 1989. [https://books.google.co.uk/books?id=Duku89bARgoC]

Reference books
*Nihon No Sanko Tosho. Volume 1: 1965. Volume 2: 1972.
**Guide to Japanese Reference Books. American Library Association. Chicago. 1966: [https://books.google.co.uk/books?id=0rflAAAAMAAJ]. Supplement. 1979: [https://books.google.co.uk/books?id=j05_F9OHzkQC]. Commentary: Encyclopedia of Library and Information Science, vol 21, [https://books.google.co.uk/books?id=H1pNvzr_n98C&amp;pg=PA149#v=onepage&amp;q&amp;f=false p 149].

Media
*[https://www.bbc.com/news/world-asia-pacific-15217593 Japan media guide]. News. BBC. 20 March 2023.
*Masaaki Kasagi. Mass Media in Japan. (Orientation seminars on Japan, number 14). 1983. [https://books.google.co.uk/books?id=odkgAAAAIAAJ]
*Routledge Handbook of Japanese Media [https://books.google.co.uk/books?id=zilKDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Publishers
*[https://www.publishersweekly.com/pw/by-topic/international/international-book-news/article/99729-get-to-know-these-japanese-publishing-companies.html Get to Know These Japanese Publishing Companies]. Publishers Weekly. 20 February 2026.

Press and journalism
*[https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2025/japan Japan]. Reuters Institute for the Study of Journalism. 17 June 2025.
*Marjane Aalam and Philippe Régnier. The Japanese Press and Information System. The Graduate Institute of International Studies. Geneva. [https://books.google.co.uk/books?id=RTcbAQAAIAAJ]
*The Japanese Press: Past and Present. Japan Newspaper Publishers' and Editors' Association. [https://books.google.co.uk/books?id=5tcQAAAAIAAJ 1949].
*Anthony Rausch. Japanese Journalism and the Japanese Newspaper: A Supplemental Reader. [https://books.google.co.uk/books?id=mZrToQEACAAJ]
*Frank L Martin. The Journalism of Japan. 1918. [https://books.google.com/books?id=ruYzAQAAMAAJ]
*William De Lange. A History of Japanese Journalism. Japan Library. 1998. [https://books.google.co.uk/books?id=Rd5tb0cuz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kanesada Hanazono. The Development of Japanese Journalism. Osaka. 1924. [https://books.google.co.uk/books?id=z99ZAAAAMAAJ]
*Kanesada Hanazono. Journalism in Japan and Its Early Pioneers. 1926. [https://books.google.co.uk/books?id=IGTFfLc4bq0C]
*César Castellvi. A Sociology of Journalism in Japan: The Last Empire of the Press. 2024. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false]
*&quot;Japan&quot;. Christopher H Sterling (ed). Encyclopedia of Journalism. A Sage Reference Publication. 2009. ISBN 9780761929574. vol 3. pp [https://books.google.co.uk/books?id=ZQhDq8fPj2IC&amp;pg=PA809#v=onepage&amp;q&amp;f=false 809] to 815.

Press annuals
*The Japanese Press. (Nihon Shinbun Kyokai). [https://books.google.co.uk/books?id=AfvyAAAAMAAJ 1979] [https://books.google.co.uk/books?id=Au3yAAAAMAAJ 1998]

Summaries of the press
*Daily Summary of Japanese Press

Foreign correspondents
*Foreign Correspondents in Japan: Reporting a Half Century of Upheavals, from 1945 to the Present. Tuttle. 1998. [https://books.google.co.uk/books?id=YI3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Nunn (comp). Japanese Periodicals and Newspapers in Western Languages: An International Union List. Mansell. 1979. [https://books.google.co.uk/books?id=jEROAQAAIAAJ]
*Japan Periodicals. Keizai Koho Center. 3rd Ed [https://books.google.co.uk/books?id=ATm0AAAAIAAJ]. Japan Periodicals, 1982. [https://books.google.co.uk/books?id=PkMyAAAAMAAJ]
*Japanese Periodicals Index
**Humanities and Social Sciences [https://books.google.co.uk/books?id=nXX_RpPGf3AC]
**Natural Sciences [https://books.google.co.uk/books?id=FCJIAAAAYAAJ]
*Current Japanese Periodicals [https://books.google.co.uk/books?id=FjO5AAAAIAAJ]
*Check-list of Japanese Periodicals Held in British University and Research Libraries. [https://books.google.co.uk/books?id=VZgsAAAAYAAJ]
*Union List of Current Japanese Periodicals in the East Asian Libraries of Columbia, Harvard, Princeton, and Yale Universities. [https://books.google.co.uk/books?id=yw7kAAAAMAAJ]
*List of Japanese Periodicals in the Library of the School of Oriental &amp; African Studies. [https://books.google.co.uk/books?id=RREjAQAAIAAJ]
*Gianni Simone. [https://www.japantimes.co.jp/community/2011/04/26/issues/english-mags-approach-milestone-crossroads/ English mags approach milestone, crossroads]. The Japan Times. 26 April 2011.
*Japan Report (1955 onwards) (Consulate General of Japan, Japan Information Center). Vol 39 published in 1993. [https://books.google.co.uk/books?id=MX4BN_frv4IC&amp;pg=PP7#v=onepage&amp;q&amp;f=false] editions:jYuMSMIQC-AC
**Japan Information
*Japan Now [https://books.google.co.uk/books?id=Nul7DRQaexMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Japan Quarterly. (Asahi Shimbun). 1954 to 2001. [https://books.google.co.uk/books?id=nZMMAQAAMAAJ] [https://books.google.co.uk/books?id=_RwVAAAAMAAJ] 189 issues.
*Japan Illustrated: The Japan Times Quarterly [Pictorial] Magazine (October 1963 to Summer 1977) 15 vols [https://books.google.co.uk/books?id=D7UThOmE8T4C]
*[[w:Japan Spotlight|Japan Spotlight]]. Economy, Culture &amp; History: Japan Spotlight: Bimonthly. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ]
*Focus Japan. (Japan External Trade Organization, JETRO). [https://books.google.co.uk/books?id=2fG2hsEZpRkC]
*The Japan Journal [https://books.google.co.uk/books?id=2V3hAAAAMAAJ] [https://books.google.co.uk/books?id=CJwoAQAAMAAJ]
*Japan Magazine. Muromachi Publicity Corporation. (vols 1 to 5: 1957 to 1963). [https://books.google.co.uk/books?id=Swd18PnVeUgC]
*The Japan Magazine: A Representative Monthly of Things Japanese [https://books.google.co.uk/books?id=ubGKo-p6O_0C] [https://archive.org/details/jm-1914-v4.9-5.2/mode/1up]
*Transactions and Proceedings of the Japan Society, London [https://books.google.co.uk/books?id=B75nnph5qHgC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
**Bulletin. [Bulletin of the Japan Society, London.] [https://books.google.co.uk/books?id=Pd9KvyhnpjMC]
**The Japan Society of London Bulletin [https://books.google.co.uk/books?id=XxlxAAAAMAAJ]
*About Japan. Japan Society, New York. [https://books.google.co.uk/books?id=Nf5OAQAAIAAJ]
**News Bulletin [https://archive.org/details/bub_gb_QcA3AQAAIAAJ/page/n2/mode/1up]
*[[w:en:Metropolis (free magazine)|Metropolis]] (metropolisjapan.com)
*[[w:en:Tokyo Weekender|Tokyo Weekender]] (トーキョー・ウィークエンダー) [https://www.tokyoweekender.com/japan-life/news-and-opinion/nhk-world-features-the-tokyo-weekender-magazine/]
*The Japan Gazette [https://books.google.co.uk/books?id=WSopAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Tokio Times [https://books.google.co.uk/books?id=UDfiFBu0vB4C&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*[[w:en:Look Japan|Look Japan]]. (Look Japan Ltd). [https://books.google.co.uk/books?id=QnO6AAAAIAAJ]. Commentary: Gale Directory of Publications and Broadcast Media [https://books.google.co.uk/books?id=ve4dAQAAMAAJ]
*[[w:en:Japan Echo|Japan Echo]]. 1974 to 2010. [https://books.google.co.uk/books?id=Cmq6AAAAIAAJ] [https://books.google.co.uk/books?id=fpmEPpl-85UC]
*PHP Intersect. (Where Japan Meets Asia and the World). PHP Institute. [https://books.google.co.uk/books?id=i74TAQAAMAAJ]
**Intersect Japan [https://books.google.co.uk/books?id=sL8TAQAAMAAJ]
*Speaking of Japan [https://books.google.co.uk/books?id=U7S0AAAAIAAJ]. [Speeches.]
*The Hansei Zasshi: A Monthly Magazine [https://books.google.co.uk/books?id=6qBhfHZo7Q0C&amp;pg=PP5#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=dyIsvnYjpwEC&amp;pg=PP6#v=onepage&amp;q&amp;f=false]
**The Orient. 1899 onwards [https://books.google.co.uk/books?id=nS1omYYnnd4C&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Today's Japan. Orient/West Incorporated. [https://books.google.co.uk/books?id=g2ASAAAAMAAJ]
*Japan Review: Bulletin of the International Research Center for Japanese Studies. [https://books.google.co.uk/books?id=GggOAQAAMAAJ]

Newspapers

See also [[w:List of newspapers in Japan]]
*Haruhara Akihiko, &quot;English-language newspapers in Japan&quot; (1994) 41 Japan Quarterly [https://www.proquest.com/openview/8e2b760f2a2fa37ba164ea675c095353/1 474] (Issue 4: October 1994)
*Tanner. English Language Newspapers in Bakumatsu Japan. 1977. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*[https://www.japantimes.co.jp/news/2009/03/03/reference/newspapers-here-soldiering-on/ Newspapers here soldiering on]. The Japan Times. 3 March 2009.

*[[w:The Japan Times|The Japan Times]]
**The Japan Times: Weekly Edition [https://books.google.co.uk/books?id=KoQ-AQAAMAAJ] [https://books.google.co.uk/books?id=yYQ-AQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Japan Daily Mail
*Japan Weekly Mail
*The Japan Chronicle
**Weekly Edition [https://books.google.co.uk/books?id=vXdRAQAAIAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Japan News. (The Japan News by The Yomiuri Shimbun)
**Yomiuri Japan News (from 1955)
**The Yomiuri (from 1958)
**The Daily Yomiuri (from 1970)
*The Asahi Shimbun: Asia &amp; Japan Watch. [https://www.asahi.com/sp/ajw/]
**Asahi Evening News (from 1954)
***Tokyo Evening News (1952 to 1954) [https://ndlsearch.ndl.go.jp/books/R100000002-I000000145073]
*The Mainichi. [https://mainichi.jp/english/]
**Mainichi Daily News (1922 to 2001) [https://www.nytimes.com/2001/02/27/business/worldbusiness/IHT-tech-briefstop-the-presses.html] [https://ndlsearch.ndl.go.jp/books/R100000002-I000000144910]

Sports newspapers; sports dailies
*Louise do Rosario, &quot;News-stand stars&quot; in &quot;Japan&quot; (1992) [https://books.google.co.uk/books?id=T_GzAAAAIAAJ 155] [[w:en:Far Eastern Economic Review|Far Eastern Economic Review]], 24 to 31 December 1992, p 21
*[[w:ja:岡崎満義|Mitsuyoshi Okazaki]], &quot;Unsportsmanlike Journalism: Japan's sports dailies may be popular, but are they sporting?&quot; in &quot;Sport&quot;, [[w:en:Look Japan|Look Japan]], [https://books.google.co.uk/books?id=lD3tAAAAMAAJ January 1995], p 39

News
*[[w:en:Japan Today|Japan Today]] (ジャパントゥデイ). GPlusMedia. Gakken Holdings.

Annuals and year books 
*This is Japan. Asahi Shimbun. 1954 to 1971. [https://books.google.co.uk/books?id=2X9DAQAAIAAJ]. Commentary: A Victorian Sailor's Grave in the Seto Inland Sea, p 244 [https://books.google.co.uk/books?id=OegkAgAAQBAJ&amp;pg=PA244#v=onepage&amp;q&amp;f=false]
*The Japan Year Book. The Japan Year Book Office. 1905 onwards. [https://archive.org/details/bub_gb_arFPAAAAMAAJ/page/n10/mode/1up 1906]. [https://archive.org/details/in.ernet.dli.2015.553496/page/n27/mode/1up 1915].
*The &quot;Japan Gazette&quot; Japan Year Book. The Japan Gazette. [https://archive.org/details/japan-year-book-1913-1914/page/n15/mode/1up 1913-14]
*The Japan Times Year Book

Almanacs
*Asahi Shimbun Japan Almanac. [https://books.google.co.uk/books?id=SEEEAQAAIAAJ 1995].
*Japan Almanac. (The Mainichi Newspapers). [https://books.google.co.uk/books?id=ufAIAQAAIAAJ 1972]. [https://books.google.co.uk/books?id=X4eXWRkbtFsC 1973]. [https://books.google.co.uk/books?id=7rMrAAAAIAAJ] [https://books.google.co.uk/books?id=krMrAAAAIAAJ]
*[[w:Boyé Lafayette De Mente|Boye De Mente]]. Passport's Japan Almanac. [https://books.google.co.uk/books?id=741wAAAAMAAJ]

General
*Japan: A Country Study. (Area Handbook series). 4th Ed: 1983: [https://books.google.co.uk/books?id=HkM5N3JNc5IC]. 5th Ed: 1992: [https://books.google.co.uk/books?id=ze-wupXxpvEC]
*Area Handbook for Japan. 2nd Ed: 1964: [https://books.google.co.uk/books?id=WucdAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]. 3rd Ed: 1974: [https://books.google.co.uk/books?id=LG2aoq1U_eoC&amp;pg=PR1#v=onepage&amp;q&amp;f=false] (DA Pam 550-30).
*Colin Simpson. Picture of Japan.
**Japan: An Intimate View. A S Barnes. [https://books.google.co.uk/books?id=3hkeAAAAMAAJ]
**This is Japan. Angus &amp; Robertson. [https://books.google.co.uk/books?id=HJEJAQAAIAAJ]
*Japan. (The World and Its Peoples). Greystone Press, New York. 1964. Volume 1: [https://books.google.co.uk/books?id=yysUAQAAMAAJ]. Volume 2 &quot;Japan Korea&quot;, including Korea: [https://books.google.co.uk/books?id=uQAUAQAAMAAJ]. See pp 1 to 375 for Japan, and pp 376 to 379 for Ryukyu and Bonin Islands.
*Japan. (World and its Peoples: Eastern and Southern Asia, volume 8). Marshall Cavendish. 2008. ISBN 9780761476412.
*Edward Seidensticker. This Country, Japan. Kodansha International. 1979. ISBN 9780870112294. [https://books.google.co.uk/books?id=88wwAQAAIAAJ]
*Hall and Beardsley. Twelve Doors to Japan. McGraw-Hill. New York. 1965. [https://books.google.co.uk/books?id=0KpxAAAAMAAJ]

Handbooks
*Heenan (ed). The Japan Handbook. (Regional Handbooks of Economic Development). 1998. [https://books.google.co.uk/books?id=IMG2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Introducing Japan Through Books: A Selected Bibliography. Public Information Bureau, Ministry of Foreign Affairs, Japan. 1968. [https://books.google.co.uk/books?id=FvsyAQAAIAAJ]. 2nd Ed: 1973: [https://books.google.co.uk/books?id=Vj0XAQAAMAAJ].
*Donald Ritchie. Introducing Japan. 1st Ed: 1978. Revised Ed: 1986. 6th printing: 1989: [https://books.google.co.uk/books?id=FE-nxxoKayQC]. 2nd Revised Ed: 1990. 2nd printing: 1991: [https://books.google.co.uk/books?id=hz4UAQAAIAAJ]. 1994: [https://books.google.co.uk/books?id=FMvT6m4SgIQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Webb. An Introduction to Japan. 2nd Ed: 1957: [https://books.google.co.uk/books?id=YQ8MAQAAIAAJ].
*Introducing Modern Japan. A publication of the Japan Information and Culture Center, Embassy of Japan.

Today and yesterday
*Ray Downs. Japan Yesterday and Today. Praeger Publishers. 1970. [https://books.google.co.uk/books?id=PwKxAAAAIAAJ]

Today
*Buckley. Japan Today. 3rd Ed [https://books.google.co.uk/books?id=thyqBtJp2DcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Contemporary
*Routledge Handbook of Contemporary Japan. 2021. [https://books.google.co.uk/books?id=yfH3DwAAQBAJ&amp;pg=PA2011#v=onepage&amp;q&amp;f=false]
*McCargo. Contemporary Japan. 3rd Ed: 2012. [https://books.google.co.uk/books?id=8I5KEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kingston. Contemporary Japan: History, Politics, and Social Change since the 1980s. [https://books.google.co.uk/books?id=enJQZA3R4FMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

[Series]
*Routledge Contemporary Japan Series

Modern
*Cortazzi. Modern Japan: A Concise Survey. 1993. [https://books.google.co.uk/books?id=Cf--DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

The Japanese
*Tasker. The Japanese: Portrait of a Nation. 1989 [https://books.google.com/books?id=Q1N8ld78wwQC]
**The Japanese: A Major Exploration of Modern Japan. [https://books.google.co.uk/books?id=CW-6AAAAIAAJ]
**Inside Japan: Wealth, Work and Power in the New Japanese Empire. 1987. [https://books.google.co.uk/books?id=2OJuAAAAMAAJ]

Travel books
*DK Eyewitness Travel: Japan. Reprinted with revisions. 2015: [https://books.google.co.uk/books?id=g2NaBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2017: [https://books.google.co.uk/books?id=vg15DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Dodd and Richmond. The Rough Guide to Japan. 2nd Ed: 2001: [https://books.google.co.uk/books?id=pRGq95ytWZoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Frommer's Japan. 5th Ed: 2000: [https://books.google.co.uk/books?id=-QC8mVyvPa8C].
*Fodor's Japan YYYY. 1984. [https://books.google.co.uk/books?id=aH2Ow27HUQ0C 1986]. [https://books.google.co.uk/books?id=3gTTf6nbv20C 1987]. 1988.
**Fodor's YY Japan. [https://books.google.co.uk/books?id=9QMHllzldlYC 91]. 92. 93.
**Fodor's Japan. 13th Ed: 1996: [https://books.google.co.uk/books?id=cZxZAAAAYAAJ]
*The New Official Guide: Japan. Japan Travel Bureau. 1966. [https://books.google.co.uk/books?id=HoxxAAAAMAAJ]
*Here is Japan. Asahi Broadcasting Corporation. [https://books.google.co.uk/books?id=8QXRCTMNG7MC]
*Japan. (Nagel Travel Guide Series, vol 32). 1964. [https://books.google.co.uk/books?id=QsbXAAAAMAAJ]
*Clark. All the Best in Japan: with Manila, Hong Kong, and Macao. (&quot;All the Best&quot; series). 1959. Reprinted 1964. [https://books.google.co.uk/books?id=yUq4YaaryrwC]. Reviews: [https://archive.dartmouthalumnimagazine.com/article/1958/6/1/all-the-best-in-japan] (1958) 110 Travel 51 [https://books.google.co.uk/books?id=UVwXAQAAMAAJ] 3 Bulletin of the Japan Society, London, No 11: June 1960, p 25 [https://books.google.co.uk/books?id=2oy74hRRXk4C]
**All the Best in Japan and the Orient. 1967.

Languages

See [[Universal Bibliography/Languages/Japanese|Japanese]]

Literature

See [[Universal Bibliography/Literature#Japanese|Japanese literature]]

Music

See [[Universal Bibliography/Music#Japanese and Japan|Music of Japan]]

Cinema

See [[Universal Bibliography/Cinema#Japanese|Cinema of Japan]]

Culture

See [[Universal Bibliography/Culture#Japanese|Culture of Japan]]

==Korea==
*Korea Journal [https://books.google.co.uk/books?id=O6XfBexsp6gC]

Bibliography and literature
*Thomas H Kang. &quot;Korean Literature and Bibliography&quot;. Kent, Lancour and Daily (eds). Encyclopedia of Library and Information Science. Marcel Dekker. 1977. vol 21. pp [https://books.google.co.uk/books?id=H1pNvzr_n98C&amp;pg=PA176#v=onepage&amp;q&amp;f=false 176] to 240.

[[Category:Countries]]</text>
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    <title>Abstract Syntax Tree</title>
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This learning resources about the generation of an abstract syntax tree from a given document. The learning resource will address the syntactical analysis MediaWiki documents, e.g. in Wikiversity. 

== History of Learning Resouce ==
Learning resource was generated in the context of passes for MarkDown in Markup languages like HTML. Abstract Syntax Trees (AST) are generated when parsers analyse the source code in a specific programming language. The tree structures represent nested environments in a programming language. As an introduction to [[w:en:Compiler|Compilers Theory]] we will create an Abstract Syntax Tree (AST) for text document.

== Decomposition of a Document ==
[[File:AST Document.svg|thumb|Abstract Syntax Tree Document]]
In a first step we decompose a document into tree nodes. A tree consists of connected tree nodes. A tree is a specific graph in which die nodes (vertices) may have attributes. In our context is 
* '''(Document Node)''' the root node of the tree is the ''&quot;document&quot;'' node. The document node contains the information about the document which was parsed. If we parse the document in Wiki markdown, we would add references and time stamps and version id of the source document, so that JSON file contains the information about the origin of the document. This can be compared with a citation in scientific paper where a certain part of the text in an article refers to source of the cited content.
* '''(Content Element List)''' is basically a array of content elements. An array is required because the order of the content elements must be preserved.
* '''(Content Element)''' Can be anything from text block to media elements like video, audio or images.
Nested structures can be created by using a &quot;Content Element List&quot; as &quot;Content Element&quot;.
** &quot;Sections&quot; a specific &quot;Content Element Lists&quot; that contain a &quot;Title&quot; and &quot;Content Element List&quot; of &quot;paragrapha&quot;,
** &quot;Paragraph&quot; 
** &quot;Images&quot;
** &quot;Audio&quot;
** &quot;Video&quot;
** &quot;Quiz&quot;
** &quot;Mathematical Expression&quot; - Type-Attribute: inline/block 
** ...

== Learning Tasks ==
* Define a JSON structure for Wikiversity Quizzes?
* Try to define a grammar and create your own parser. A good starting point is a&lt;code&gt;PRODUCT&lt;/code&gt; grammar for basic mathematical expressions like &lt;code&gt;(4+3)*(8/2)&lt;/code&gt; and decompose that into tree. Root node of the tree will be for example a &lt;code&gt;PRODUCT&lt;/code&gt; token, that decomposes in to child nodes that are
** an &lt;code&gt;ADDITION&lt;/code&gt; and
** a &lt;code&gt;FRACTION&lt;/code&gt;
** the addition and fractions decompose into two numbers each. To learn about grammars and parsers you might want to start with online experiments with [https://pegjs.org/online PEG.JS]&lt;ref&gt;Wakita, K., Homizu, K., &amp; Sasaki, A. (2014, August). Hygienic macro system for JavaScript and Its light-weight implementation framework. In Proceedings of ILC 2014 on 8th International Lisp Conference (pp. 12-21).&lt;/ref&gt; 
* How can you export a Wiki source for a quiz from a JSON?
* What are the benefits and drawbacks of a JSON format for a Wikiversity quiz instead of having the source code available directly as a text file?
* Analyze the concept of a [[JSON Editor]] and create Quiz editor that is able to generate 
** a quiz for Wikiversity and
** a quiz for the [https://github.com/surveyjs/survey-library OpenSource library SurveyJS]&lt;ref&gt;SurveyJS GitHub Respository (2021) URL: https://github.com/surveyjs/survey-library (Accessed 2021/01/22)&lt;/ref&gt;
* '''([[Parser Generator]])''' A parser generator gets as input a [[w:en:Grammar|Grammar]] and returns parser ready to use for the defined grammar. 
** Use the Open Source parser generator [https://pegjs.org/online PEG] and create a parser in Javascript that decompose the HTML text into the sections. 
** Create a JSON of the following structure 
   [
       {
          title:&quot;Introduction&quot;,
          level: 1,
          text: &quot;My text of the introducton&quot;  
       },
       {
          title:&quot;Definition&quot;,
          level: 1,
          text: &quot;provide a definition of the topic.&quot;  
       }

   ]

== See also ==
* [[Natural Language Processing]]
* [[w:en:Parsing|Parser]]
* [[w:en:Compiler#Compiler_construction|Compiler Theory]]
* [[PanDocElectron]]
* [[JSON Editor]]
* [https://www.github.com/niebert/XML2Code XML2Code]

== References ==

[[Category:Grammar]]
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    <title>Universal Bibliography/Law/Legislation</title>
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This page is part of a pan-jurisdictional [[Universal Bibliography/Law|bibliography of law]]. This part of the [[Universal Bibliography]] is a bibliography of legislation.

'''Collections of statutes; tables and idexes'''
*[[w:Halsbury's Statutes|Halsbury's Statutes]]

*[[w:Revised edition of the statutes|Revised edition of the statutes]]
**The Statutes: Revised Edition
**The Statutes: Second Revised Edition
**The Statutes Revised: Third Edition
**[[w:Statutes in Force|Statutes in Force]]
**The Acts of the Parliaments of Scotland 1424-1707: Revised Edition. 1908. [https://books.google.co.uk/books?id=hGk2AAAAIAAJ]. Second Revised Edition. 1966.
**[[w:The Northern Ireland Statutes Revised|The Northern Ireland Statutes Revised]]

*[[w:Legislation.gov.uk|Legislation.gov.uk]]/SLD ([http://www.legislation.gov.uk/ Legislation.gov.uk])

*[[w:Current Law Statutes Annotated|Current Law Statutes Annotated]]
*[[w:Annotated Legislation Service|Annotated Legislation Service]]

*[[w:Chronological Table of the Statutes|Chronological Table of the Statutes]]
*[[s:Index to the Statutes|Index to the Statutes]]

*[[w:Chronological Table of Local Acts|Chronological Table of Local Acts]]
*Index to Local Acts. 1900. [https://books.google.co.uk/books?id=9iw1AQAAMAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*[[w:Chronological Table of Private and Personal Acts|Chronological Table of Private and Personal Acts]]

*The &quot;Alphabetical Table of Statutes&quot; (from 1707 onwards) in certain volumes of Current Law Statutes

*[[w:The Statutes of the Realm|The Statutes of the Realm]]
*[[w:Statutes at Large|Statutes at Large]] [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA661#v=onepage&amp;q&amp;f=false]
**Keble [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA433#v=onepage&amp;q&amp;f=false]
**Hawkins [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA375#v=onepage&amp;q&amp;f=false]
**Cay (1751) [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA180#v=onepage&amp;q&amp;f=false]
**Ruffhead (1763) [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA622#v=onepage&amp;q&amp;f=false], Runnington (1786) [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA623#v=onepage&amp;q&amp;f=false], Raithby, Rickards
**Pickering [http://books.google.co.uk/books?id=DSNEAAAAYAAJ&amp;pg=PA571#v=onepage&amp;q&amp;f=false]

*[[s:The Public General Acts and Measures|The Public General Acts and Measures]]
*The Law Reports - Public General Statutes

*[[w:Halsbury's Statutory Instruments|Halsbury's Statutory Instruments]]
*Statutory Rules and Orders Revised. 1889.
*Statutory Rules and Orders Revised. Second Edition. 1903.
*The Statutory Rules and Orders and Statutory Instruments Revised. Third Edition. Revised to 1948.
*[[s:Table of Government Orders|Table of Government Orders]]
*[[s:Index to Government Orders|Index to Government Orders]]

*Blackstone's Guides. (Blackstone's Guide to the . . . Act YYYY).
*Blackstone's Statutes (formerly Blackstone's Statute Books)
*Routledge Student Statutes
*Butterworth's Handbooks
*Butterworth's Twentieth Century Statutes (continuation of Butterworth's Statutes) - 30 LQR 248
*Chitty's Statutes of Practical Utility - 30 LQR 118
*Chitty's Annual Statutes. (Chitty's Annual Statutes and Statutory Rules and Orders). (Chitty's Annual Statutes of Practical Utility). (The Annual Statutes of Practical Utility). Reviews: [https://books.google.co.uk/books?id=FNg2AQAAMAAJ] [https://books.google.co.uk/books?id=Mf9BAQAAMAAJ]
**Statutes of Practical Utility passed in YYYY [https://books.google.co.uk/books?id=RTw1AQAAMAAJ&amp;pg=PA591#v=onepage&amp;q&amp;f=false]
**Chitty's Statutes of Practical Utility . . . [Supplement] containing Statutes of Practical Utility passed in YYYY [https://books.google.co.uk/books?id=wepCAQAAMAAJ] [https://books.google.co.uk/books?id=h-xCAQAAMAAJ]. Reviews and commentary: [https://books.google.co.uk/books?id=kqguAAAAIAAJ] [https://books.google.co.uk/books?id=85gvAQAAIAAJ] [https://books.google.co.uk/books?id=aoRQAQAAIAAJ] [https://books.google.co.uk/books?id=UgE0AQAAMAAJ] [https://books.google.co.uk/books?id=koMzAQAAMAAJ] [https://books.google.co.uk/books?id=UvguAAAAIAAJ] [https://books.google.co.uk/books?id=4NwwAQAAIAAJ] [https://books.google.co.uk/books?id=vtc2AQAAMAAJ] [https://books.google.co.uk/books?id=11owAQAAMAAJ]
*[[w:Paterson's Practical Statutes|Paterson's Practical Statutes]].
*The Annotated Acts. Sweet &amp; Maxwell. Reviews: [https://books.google.co.uk/books?id=UVhNAQAAMAAJ&amp;pg=PA764#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=UVhNAQAAMAAJ&amp;pg=PA254#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=AwZRAQAAIAAJ&amp;pg=PA111#v=onepage&amp;q&amp;f=false]
*Williams. Everyday Statutes Annotated. Commentary: [https://books.google.co.uk/books?id=QPQ-AAAAIAAJ]
*Haynes. The Students Statutes. [https://books.google.co.uk/books?id=XtsDAAAAQAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. 3rd Ed: 1884 [https://books.google.co.uk/books?id=Ko4DAAAAQAAJ&amp;pg=PP3#v=onepage&amp;q&amp;f=false]
*Weekly Reporter Statutes: [https://books.google.co.uk/books?id=j4JhAAAAcAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false] Reviews: [https://books.google.co.uk/books?id=LHJGAQAAIAAJ&amp;pg=PA272#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=NfM5AQAAMAAJ&amp;pg=PA992#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=NzRMAQAAIAAJ&amp;pg=RA2-PA171#v=onepage&amp;q&amp;f=false]. [https://books.google.co.uk/books?id=7Q8vAAAAIAAJ&amp;pg=PA211#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=lRU-AQAAIAAJ&amp;pg=PA1218#v=onepage&amp;q&amp;f=false]. See [[w:John Warrington Rogers]].
*Keen, Local Legislation, 1913 - 30 LQR 514

*The Scots Statutes Revised. William Green. The Public General Statutes Affecting Scotland. [https://archive.org/details/scotsstatutesre00britgoog v7], [https://archive.org/details/scotsstatutesre00unkngoog v9]. The Acts of the Parliaments of Scotland 1424-1707: [https://archive.org/details/actsparliaments00scotgoog]
*The Scots Statutes. William Green
*The Public General Statutes affecting Scotland. (A Collection of the Public General Statutes affecting Scotland). William Blackwood and Sons.
*The Acts of the Parliaments of Scotland 1424-1707. [https://books.google.co.uk/books?id=F6E0AQAAMAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=HfUmgBAzjccC&amp;pg=PP9#v=onepage&amp;q&amp;f=false].
*&quot;The Scots Laws and Acts of Parliament . . .&quot; [from dedication page]. 1681. [[s:Index:The Laws and Acts of Parliament of Scotland.djvu|(wikisource)]]

*Darby. Handy Book of post-Union Statutes. 1872. Review: [https://books.google.co.uk/books?id=1tcQAAAAYAAJ&amp;pg=PA616#v=onepage&amp;q&amp;f=false]

*[[w:Irish Statute Book|The Irish Statute Book (irishstatutebook.ie)]]
*[http://www.qub.ac.uk/ild/?func=help&amp;section=acknowledgments The Irish Legislation Database]

*Irish Current Law Statutes Annotated (1984 onwards)
*Butterworths Irish Annotated Statutes

*Acts of the Oireachtas as Promulgated [https://books.google.co.uk/books?id=Pd1LAQAAIAAJ] editions:IGC_LLhP9k0C
*The Public General Acts Passed by the Oireachtas of Saorstát Eireann [https://books.google.co.uk/books?id=GqovAQAAMAAJ] editions:9uGAwaL3oy4C editions:oeK_mEAPtEMC editions:EBYhYxPYHVAC

South Africa
*Juta's Statutes of South Africa
*Statutes of the Republic of South Africa: Classified and Annotated from 1910. Butterworths. Durban. 1967 onwards. Looseleaf.
*The Union Statutes: Classified and Annotated Reprint: 1910–1947. Butterworth. Durban. 1948 to 1952.
**Cumulative Supplement 
*Statutes of the Republic of South Africa, YYYY. Government Printer. [https://books.google.co.uk/books?id=1hszAAAAIAAJ]
*Statutes of the Union of South Africa, YYYY. Government Printer.
*Juta's Provincial Ordinances
*Cape of Good Hope: Acts of Parliament. [https://books.google.co.uk/books?id=_eMqAAAAYAAJ&amp;pg=PA3575#v=onepage&amp;q&amp;f=false]
*Jackson (ed). Statutes of the Cape of Good Hope, 1652-1905. [https://books.google.co.uk/books?id=AYBDAQAAMAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Tennant and Jackson (eds). Statutes of the Cape of Good Hope, 1652-1895. [https://books.google.co.uk/books?id=0m9FAQAAMAAJ&amp;pg=PA2407#v=onepage&amp;q&amp;f=false]
*Foster, Tennant and Jackson (eds). Statutes of the Cape of Good Hope, 1652-1886. Review: [https://books.google.co.uk/books?id=qZEzAQAAMAAJ&amp;pg=PA30#v=onepage&amp;q&amp;f=false]
*Statutes of the Cape of Good Hope passed by the Xth Parliament during the Sessions YYYY-YYYY
*A Selection of the Statutes of the Cape of Good Hope as Set by the University of Cape of Good Hope for the Law Examinations
*Statute Law of the Cape of Good Hope [https://books.google.co.uk/books?id=QOVRAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Tennant (compiler). Alphabetical Index to the Statute Law of the Colony of the Cape of Good Hope. 1877. [https://books.google.co.uk/books?id=5XtDAQAAMAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Ordinances of the Province of the Cape of Good Hope
*Acts of the Parliament of the Colony of Natal
*Statutes of Natal
*Natal Ordinances, Laws, and Proclamations
*Ordinances, Proclamations, &amp;c, &amp;c, &amp;c, relating to the Colony of Natal
*Ordinances of the Province of Natal
*William Broome. The Laws of Natal.
*Statutes of the Orange River Colony
*The Statute Law of the Orange River Colony
*Ordinances of the Province of the Orange Free State
*Statutes of the Transvaal
*Statute Law of the Transvaal
*Ordinances of the Transvaal
*Ordinances of the Transvaal Colony
*Statutory Proclamations of the Transvaal

France
*[[w:Légifrance|Légifrance]]

Many law books contain a table of statutes referred to in the volume.

'''Practical editions, practical statutes, statutes of practical utility'''

*Brunner. &quot;Practical Editions&quot;. The Sources of the Law of England. Translated by Hastie. 1888. [https://books.google.co.uk/books?id=_SRDOi1PcGQC&amp;pg=PA34#v=onepage&amp;q&amp;f=false p 34]

United Kingdom

See the entries for Chitty and Paterson above. The Law Times published a series of practical editions of important statutes: &quot;Notices of New Law Books&quot; (1846) 8 The Law Times [https://books.google.co.uk/books?id=_SRDOi1PcGQC&amp;pg=PA34#v=onepage&amp;q&amp;f=false 29].

Ontario
*Bicknell and Kappele. Practical Statutes [being a Collection of Statutes of Practical Utility in Force in Ontario with Notes on the Construction and Operation thereof]. Goodwin &amp; Co. Toronto. 1900. [https://archive.org/details/practicalstatute00bickuoft] [https://books.google.co.uk/books?id=kkwtAQAAMAAJ].

New Zealand
*Wilson, Walter Munro. The Practical Statutes of New Zealand. Wayte and Batger. Auckland. 1867. [https://books.google.co.uk/books?id=EXEOAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. 
*Barton's Practical Statutes. [[w:George Burnett Barton|Barton, George Burnett]]. The Practical Statutes of New Zealand. Dunedin. 1876 to 1877. [https://books.google.co.uk/books?id=WlcrAAAAYAAJ&amp;pg=PA331#v=onepage&amp;q&amp;f=false catalogue]. Commentary: [https://books.google.co.uk/books?id=neoLAQAAIAAJ&amp;pg=PA659#v=onepage&amp;q&amp;f=false]. For Barton, see [https://books.google.co.uk/books?id=02waAAAAMAAJ&amp;pg=PA104#v=onepage&amp;q&amp;f=false Allibone].

Victoria
*The Public General Statutes of Victoria of Practical Utility. editions:cHZXosLSwgwC.

New South Wales
*Oliver, Alexander. A Collection of the Statutes of Practical Utility, Colonial and Imperial, in Force in New South Wales. Thomas Richards. Sydney. 1879. [https://books.google.co.uk/books?id=D2xGAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false v1], [https://books.google.co.uk/books?id=4WNGAAAAYAAJ&amp;pg=PA1287#v=onepage&amp;q&amp;f=false v2].
*Cary, Henry (ed). A Collection of Statutes affecting New South Wales; containing all the Statutes of Practical Utility to the Present Time. Sands and Kenny. Sydney and Melbourne. 1861. [https://books.google.co.uk/books?id=AONCAQAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false v1], [https://books.google.co.uk/books?id=iuFCAQAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false v2].
*Tarleton, W W. A Collection of the Private Acts of Practical Utility in force in New South Wales; embracing the Local Private Legislation. Sydney. 1886. Catalogues: [https://books.google.co.uk/books?id=tlVBAQAAMAAJ&amp;pg=PA526#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=8GYhAQAAMAAJ&amp;pg=PA382#v=onepage&amp;q&amp;f=false]

'''Abstracts'''
*A Compendious Abstract of the Public General Acts of the United Kingdom of Great Britain and Ireland. (Law Journal Reports). [https://books.google.co.uk/books?id=_hBDAQAAMAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]. editions:fBGGcgLwSwcC editions:-3mv3mL5IBEC editions:9iA3vgspwG8C
*Williams, Thomas Walter. An Accurate Abstract of the Public General Statutes passed in ... (with notes and comments): editions:zW3zdZkcMNYC editions:uD0US8VHjXAC [https://books.google.co.uk/books?id=a3ZjAAAAcAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=DnZjAAAAcAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-3VjAAAAcAAJ&amp;pg=PP3#v=onepage&amp;q&amp;f=false] (also called &quot;Statutes of the Year&quot;: [https://books.google.co.uk/books?id=k5UwAAAAYAAJ&amp;pg=PA616#v=onepage&amp;q&amp;f=false]). A Compendious Abstract of the Public General Acts Passed in . . . [https://books.google.co.uk/books?id=YndjAAAAcAAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false]

'''Legislation of the year or session, new legislation: abstracts, epitomes, notes etc in periodicals and year books'''

*&quot;Legislation of the Year&quot;. Solicitors' Journal. [https://books.google.co.uk/books?id=ZAo5AQAAMAAJ&amp;pg=PA949#v=onepage&amp;q&amp;f=false]. &quot;A Reading of the New Statutes&quot;.
*&quot;New Laws of the Session&quot;. The Law Times: [https://books.google.co.uk/books?id=Mp4DAAAAQAAJ&amp;pg=PA301#v=onepage&amp;q&amp;f=false]. Law Times Editions of Important Statutes: [https://books.google.co.uk/books?id=8qsaAAAAYAAJ&amp;pg=RA1-PA1#v=onepage&amp;q&amp;f=false].
*&quot;New Statutes&quot; in  &quot;The Legislator&quot;. The Law Times. [https://books.google.co.uk/books?id=BuEwAQAAIAAJ&amp;pg=RA1-PA111#v=onepage&amp;q&amp;f=false].
*&quot;The Statutes YYYY-YY&quot;. The Law Times. [https://books.google.co.uk/books?id=yP0wAQAAIAAJ].
*&quot;Abstract of Public General Statutes&quot;. The Law Magazine [https://books.google.co.uk/books?id=sb1FAQAAMAAJ&amp;pg=PA487#v=onepage&amp;q&amp;f=false]
*Notes on New Statutes. The Legal Observer. [https://books.google.co.uk/books?id=T_0uAAAAIAAJ&amp;pg=PP15#v=onepage&amp;q&amp;f=false]. New Statutes. [https://books.google.co.uk/books?id=DwIvAAAAIAAJ&amp;pg=PA9#v=onepage&amp;q&amp;f=false].
*Recent Statutes, X and Y Victoria. The Law Chronicle. [https://books.google.co.uk/books?id=i3JGAQAAIAAJ&amp;pg=PA25#v=onepage&amp;q&amp;f=false]
*&quot;The Past Session&quot;. The Justice of the Peace. [https://books.google.co.uk/books?id=kGwzAQAAMAAJ&amp;pg=PA562#v=onepage&amp;q&amp;f=false]. &quot;The Statutes of the Session&quot; [https://books.google.co.uk/books?id=EW0zAQAAMAAJ&amp;pg=PA546#v=onepage&amp;q&amp;f=false]. &quot;The Statutes of the past Session&quot; [https://books.google.co.uk/books?id=GwQ5AQAAMAAJ&amp;pg=PA562#v=onepage&amp;q&amp;f=false]
*Thwaites, Charles. &quot;Epitome of Statutes&quot;. The Law Students' Journal. [https://books.google.co.uk/books?id=RwFCAQAAMAAJ&amp;pg=PA213#v=onepage&amp;q&amp;f=false].
*The Statutes X and Y Victoriae. The Journal of Jurisprudence. [https://books.google.co.uk/books?id=abNGAQAAIAAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false].
*&quot;The Legislation of YYYY&quot;. Irish Law Times. [https://books.google.co.uk/books?id=wdYQAAAAYAAJ&amp;pg=PA569#v=onepage&amp;q&amp;f=false]. &quot;The Statutes of X and Y Victoria&quot; [https://books.google.co.uk/books?id=cIwvAQAAIAAJ&amp;pg=PA497#v=onepage&amp;q&amp;f=false].
*The Statutes of YYYY. The Constitutional Year Book. [https://books.google.co.uk/books?id=JSMEAAAAYAAJ&amp;pg=PA300#v=onepage&amp;q&amp;f=false].
*&quot;Abstracts of Important Public Acts&quot;. Companion to the Almanac; or, Year-Book of General Information. (The British Almanac of the Society for the Diffusion of Useful Knowledge). Charles Knight. [https://books.google.co.uk/books?id=2jRNAAAAcAAJ&amp;pg=RA1-PA111#v=onepage&amp;q&amp;f=false].
*&quot;Review of the Legislation of the British Empire in YYYY&quot;. Journal of the Society of Comparative Legislation. [https://books.google.co.uk/books?id=5UYBAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false] &quot;Notes of the State of Legislation in America in YYYY&quot; [https://books.google.co.uk/books?id=5UYBAAAAYAAJ&amp;pg=PA232#v=onepage&amp;q&amp;f=false]. &quot;French Legislation in YYYY&quot;. &quot;German Legislation in YYYY. [https://books.google.co.uk/books?id=CCsZAAAAYAAJ 1900]. &quot;Review of Legislation YYYY&quot;. [https://cors.archive.org/details/journalsocietyc00unkngoog/page/52/mode/1up?view=theater].
*&quot;New Acts of Parliament&quot;. Public Health. [https://books.google.co.uk/books?id=EPsNAAAAYAAJ&amp;pg=PA387#v=onepage&amp;q&amp;f=false]

'''Yearbooks'''
*Yearbook of Legislation. New York State Library. [https://cors.archive.org/details/yearbooklegisla00whitgoog]

'''Chronological tables'''
*Crabb, George. A Digest and Index with Chronological Tables of All the Statutes. A Maxwell. 1841. [https://books.google.co.uk/books?id=Rk8DAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Upper Canada
*Wicksteed. Table of the Provincial Statutes in Force or which have been in Force in Upper Canada in their Chronological Order. 1856. [https://books.google.co.uk/books?id=RwIQAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

New South Wales
*Oliver, Alexander. A Chronological Table of, and General Index to, the Colonial Statutes in Force in New South Wales. Thomas Richards. Sydney. 1881. [https://books.google.co.uk/books?id=qgpLAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]

New Zealand
* &quot;Chronological Table of the Ordinances of the General Legislative Council of New Zealand&quot; [https://books.google.co.uk/books?id=NkhRAQAAMAAJ&amp;pg=RA4-PA3#v=onepage&amp;q&amp;f=false]

India
*Chronological Tables of the Indian Statutes [https://books.google.co.uk/books?id=MWHeNNl3vCcC]
*Chronological Tables and Index of the Indian Statutes [https://books.google.co.uk/books?id=ONoyAQAAMAAJ]
*Field, Charles Dickinson. Chronological Table of, and Index to, the Indian Statute-Book.1870 [https://books.google.co.uk/books?id=zXUZAAAAYAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

Pennsylvania
*Chronological Table of Statutes: Annotated 1874 to 1921 [https://books.google.co.uk/books?id=CsQ4AQAAMAAJ]

New York
*Birdseye, Clarence Frank. A Table, Chronologically Arranged, of the Statutes of the State of New York. 1887. [https://books.google.co.uk/books?id=s344AAAAIAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]. Supplement. 1894. [https://books.google.co.uk/books?id=aH44AAAAIAAJ&amp;pg=PP15#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=3WASAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=2YNZAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

'''Indexes'''
*&quot;Indexes to the Statutes&quot; [https://books.google.co.uk/books?id=Ab1IAQAAIAAJ&amp;pg=PA97#v=onepage&amp;q&amp;f=false]. Wheatley. What is an Index?. 2nd Ed: 1879.

'''Google Books and Internet Archive'''

*[http://www.google.com/search?q=editions%3Au9hJKEhmki4C&amp;tbm=bks&amp;tbo=1 editions:u9hJKEhmki4C]
*[http://www.google.com/search?q=editions%3AGegeOAOti2MC&amp;tbm=bks&amp;tbo=1 editions:GegeOAOti2MC]
*[http://www.google.com/search?q=editions%3AlDbSSa-hjKUC&amp;tbm=bks&amp;tbo=1 editions:lDbSSa-hjKUC]
*[http://www.google.com/search?q=editions%3ATmhdu8OpbqgC&amp;tbm=bks&amp;tbo=1 editions:Tmhdu8OpbqgC]
*[http://www.google.com/search?q=editions%3AUJTMp97V6hcC&amp;tbm=bks&amp;tbo=1 editions:UJTMp97V6hcC]
*[http://www.google.com/search?q=editions%3AnjO3l3LDB-YC&amp;tbm=bks&amp;tbo=1 editions:njO3l3LDB-YC]
*[http://books.google.co.uk/books?q=editions:STANFORD36105062767277 editions:STANFORD36105062767277]
*[http://books.google.co.uk/books?q=editions:STANFORD36105062766048 editions:STANFORD36105062766048]
*[http://www.google.com/search?q=editions%3A_LE0eKonAV4C&amp;tbm=bks&amp;tbo=1 editions:_LE0eKonAV4C]
*editions:T74ladB1qTAC

*Archive.org [http://www.archive.org/search.php?query=statutes][http://www.archive.org/search.php?query=The%20statutes][http://www.archive.org/search.php?query=Statutes%20revised][http://www.archive.org/search.php?query=Public%20general%20statutes][http://www.archive.org/search.php?query=Public%20general%20Acts]
*[http://www.archive.org/details/publicgeneralst01britgoog 1870], [http://www.archive.org/details/publicgeneralst00britgoog 1872], [http://www.archive.org/details/publicgeneralst02walegoog 1874], [http://www.archive.org/details/publicgeneralst00walegoog 1877], [http://www.archive.org/details/publicgeneralst05walegoog PGS 1881], [http://www.archive.org/details/publicgeneralac00unkngoog PGA 1881], [http://www.archive.org/details/publicgeneralac00walegoog 1882], [http://www.archive.org/details/publicgeneralac01walegoog 1884], [http://www.archive.org/details/publicgeneralst06walegoog 1892], [http://www.archive.org/details/publicgeneralst00unkngoog 1896], [http://www.archive.org/details/publicgeneralst03walegoog 1902], [http://www.archive.org/details/publicgeneralst04walegoog 1904], [http://www.archive.org/details/publicgeneralst01walegoog 1906]

Statutes Revised
*1st Ed
**editions:SzKZkY0BpRgC
**editions:3A-3PReZtRoC
*2nd Ed
**editions:ELugFCjXmwgC
**editions:Qo-nSwOZJZMC
**editions:rJI9AQWoE0cC
*3rd Ed
**editions:G0CJV-h2HdMC

The Acts of the Parliaments of Scotland: Revised Edition
*editions:bMkyQGghfQYC

Paterson's Practical Statutes [https://archive.org/details/practicalstatut00pategoog 1902], [https://archive.org/details/practicalstatut03pategoog 1903]
*[https://books.google.co.uk/books?id=mTw1AQAAMAAJ] [https://books.google.co.uk/books?id=wTw1AQAAMAAJ]
*editions:y0sp-_2Gl5kC
*editions:G_BbpMqfiwAC
*editions:iE3SfyRQnnIC

Chitty's Statutes
*editions:9sLvgCO6-kYC
*editions:6GH9EMbBqE4C
*editions:ofjpw4aqW94C
*editions:77BSVbe0vL0C
*editions:73itgjVJtzQC
*editions:dO_by-sGUGMC
*Annual
**editions:pmcJb8OZddMC
**editions:sUkk4hRsEhoC

A Collection of the Public General Statutes
*editions:XqtIWArt7EQC
*editions:rp1OvZ3x9HAC
*editions:_LE0eKonAV4C
*editions:aljEcqeh0DMC
*editions:TfK_BN_vSBcC
*editions:nD9gEtWfSQ4C

The Acts of the Parliaments of Scotland
*editions:0dzoBz5FVe0C

Scots Statutes
*editions:ywfbc5p8UBAC

Scots Statutes Revised
*editions:AgtkC6ydi-8C

Public Bills

*editions:Audmdw1ZNhcC
*editions:DQFPF3UM8rAC
*editions:tkCnz5fHQqkC
*editions:PsxY3j7rEUgC
*editions:4ZMw5RMM5BgC

Local and Personal Acts, The Local Acts, The Private Acts
* editions:RJQ4yovyLJwC
* editions:Wv11ZEky6Q0C

'''Other works'''
*Bennion. Statute Law.
*Bennion. Statutory Interpretation.
*Miers and Page. Legislation.
*Craies on Statute Law. [http://archive.org/details/cu31924021652361 1911]. (formerly Hardcastle on Statutory Law). [http://archive.org/details/atreatiseoncons00craigoog 1892]. Now Craies on Legislation.
*Maxwell on Statutes. (On the Interpretation of Statutes). (The Interpretation of Statutes). 1st Ed: 1875. 2nd Ed: 1883. 3rd Ed: 1896. 4th Ed: 1905. 5th Ed: 1912. 6th Ed: 1920. 7th Ed: 1929. 8th Ed: 1937. 9th Ed: 1946. 10th Ed: 1953. 11th Ed: 1962. 12th Ed: 1969.
*Cross on Statutory Interpretation
*The Law Commission. [http://www.bailii.org/ew/other/EWLC/1969/21.html The Interpretation of Statutes]. Law Com 21. 1969.
*Ilbert. [[w:Legislative Methods and Forms|Legislative Methods and Forms]]. Oxford. 1901.
*Evans. Statutory Interpretation.
*Lowe and Potter. Understanding Legislation: A Practical Guide to Statutory Interpretation. [https://books.google.co.uk/books?id=rJNVDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Dale. Legislative Drafting: A New Approach.

Europe
*Karpen and Xanthaki (eds). Legislation in Europe. Hart Publishing. 1st vol: 2017: [https://books.google.co.uk/books?id=PkobDgAAQBAJ&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd vol: 2020: [https://books.google.co.uk/books?id=2S8LEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Ireland
*Hunt, Brian. The Irish Statute Book: A Guide to Irish Legislation. First Law Limited. 2007. [https://books.google.co.uk/books?id=giBMAQAAIAAJ] (consultant ed: Francis Bennion)
*Vance. The Irish Statutes. 1862 [https://books.google.co.uk/books?id=pPYZAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. The Green Book; or, Reading Made Easy of the Irish Statutes. 1862. [https://books.google.co.uk/books?id=WfMZAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=JXyr-xDHvv8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false] (digest of case law on interpretation of particular Acts)
*Hand, Geoffrey Joseph. &quot;Statutes&quot;. English Law in Ireland 1290-1324. Cambridge University Press. 1967. Chapter 8. [https://books.google.co.uk/books?id=yzU8AAAAIAAJ&amp;pg=PA159#v=onepage&amp;q&amp;f=false Page 159] et seq.

'''Countries'''

New Zealand

See [[Universal Bibliography/Law/New Zealand|Legislation of New Zealand]]

'''Bills'''

Bills in US Congress
*[[w:THOMAS|THOMAS]]
*[[w:GovTrack|GovTrack]]

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      <text bytes="112011" sha1="m9dzm5px74nnogg7if1qhwn3r8f9ju2" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>m9dzm5px74nnogg7if1qhwn3r8f9ju2</sha1>
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    <revision>
      <id>2830537</id>
      <parentid>2830536</parentid>
      <timestamp>2026-09-02T16:21:33Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="112016" sha1="1b0njjqh717skq1y4fq5yuvxf3ioxgf" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="112004" sha1="6uyb05o4sofvez7sy8hh0c7bxpdclan" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|first imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="111998" sha1="cux0vfmxo75cpb7mftuqbz5qh1mwd50" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2830542</id>
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      <timestamp>2026-09-02T18:05:11Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* Origins of the theory */</comment>
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      <text bytes="111998" sha1="tu1exucwq8a2hmw51e8uz20689pxp0a" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="112022" sha1="ngqvgnqdckto50t3gbclk6lr32iazld" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined motions and objects. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="112022" sha1="smulc2ngvl9x3g12y6cvtug1zxaulbl" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="112032" sha1="r52zvzmn9d92vso0bi2mnvlckn9g0k1" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="112030" sha1="fd5bbgl9wmnktlat64n0civmur7dv5v" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli and Coxeter's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="112018" sha1="nb32r92s74vobwlilbe4p4x5xyywu9j" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions, in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="112015" sha1="k3wmo7zmrndrl47n1nq0bqw1jx9j4xi" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover in imagination and then explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="112025" sha1="dt1cc401evop28es6jo2otjwzsbtao0" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. This project is forever beginning anew. Coxeter showed that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="112025" sha1="dtp1cde1z4wmpxg94nee1w9uhajk5af" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter showed that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="112033" sha1="p3t1mktft9j1jyl27gyjmaugxlwtqju" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="112043" sha1="2hzkfdotolr6erczkh7p5fjl12vm3l4" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...


== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <comment>/* Distribution of stars in our galaxy */</comment>
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      <text bytes="112051" sha1="023to3azf1ygcpl587wh8o4h1ff3qbs" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

{{Efn|
== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="112049" sha1="br4zs3g1nahv5etc44czwbwa1glbguj" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="113368" sha1="fc5rhonbwtqzlnqnrse84ohwtkmvzb7" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To plot the galaxy's stars in 4-coordinate space so we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space in our fourth dimension. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="113379" sha1="lfww2k9tjq7he3awv8o66e0qnftxtpc" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To plot the galaxy's stars in 4-coordinate space so we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space in our fourth dimension. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>lfww2k9tjq7he3awv8o66e0qnftxtpc</sha1>
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    <revision>
      <id>2830602</id>
      <parentid>2830601</parentid>
      <timestamp>2026-09-02T23:28:35Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Distribution of stars in our galaxy */</comment>
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      <text bytes="113376" sha1="55zhwvwz7xloz1g7ku9cvj5v1nx8p7s" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To plot the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space in our fourth dimension. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="113384" sha1="c5n8vv3qjwwp35fsh66kjj2m1mxmt5x" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To plot the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space in our fourth dimension. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="113360" sha1="n14y0l80nq7ah51fjhriypyf16ndyen" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To plot the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>n14y0l80nq7ah51fjhriypyf16ndyen</sha1>
    </revision>
    <revision>
      <id>2830605</id>
      <parentid>2830604</parentid>
      <timestamp>2026-09-02T23:39:56Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Distribution of stars in our galaxy */</comment>
      <origin>2830605</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="113359" sha1="6vqce6542b78x6jdhtlim88gk08chi7" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>6vqce6542b78x6jdhtlim88gk08chi7</sha1>
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    <revision>
      <id>2830606</id>
      <parentid>2830605</parentid>
      <timestamp>2026-09-02T23:45:35Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Distribution of stars in our galaxy */</comment>
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      <text bytes="113366" sha1="8t6s1nlep4hd91i6kybynnk24s9b9yy" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="113366" sha1="6juj04sllgaj2jg32gtfk2qa6lqsfad" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="113373" sha1="rcl9158ss3ufjsfnbukdy41dlz7film" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently nearly directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>rcl9158ss3ufjsfnbukdy41dlz7film</sha1>
    </revision>
    <revision>
      <id>2830610</id>
      <parentid>2830609</parentid>
      <timestamp>2026-09-02T23:54:34Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Distribution of stars in our galaxy */</comment>
      <origin>2830610</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="113366" sha1="6juj04sllgaj2jg32gtfk2qa6lqsfad" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a persuasive thought experiment, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* Distribution of stars in our galaxy */</comment>
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      <text bytes="113392" sha1="3dnzzm8p27qypr1eu94q8h4b1ogcms1" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.

...have to perform this experiment somehow, before I publish this paper...
}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="113315" sha1="7oo0olxxyp7deprz14sm5j8mdnu3nlw" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="113329" sha1="4m3zyylyymeep9boz0z3x9114qoxo2u" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their red-shift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>4m3zyylyymeep9boz0z3x9114qoxo2u</sha1>
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    <revision>
      <id>2830614</id>
      <parentid>2830613</parentid>
      <timestamp>2026-09-03T00:06:53Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Light and Mass are Reflection and Rotation */</comment>
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      <text bytes="113328" sha1="hfzaz7v911njzrv47f9nlcm4bb52ism" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2830615</id>
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      <timestamp>2026-09-03T00:11:40Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Distribution of stars in our galaxy */</comment>
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      <text bytes="113336" sha1="fyaam0btnu83rsdmm96bve5cpfy4cvc" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see exactly how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="113328" sha1="hfzaz7v911njzrv47f9nlcm4bb52ism" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
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      <text bytes="113725" sha1="fieaxmytdpznbcphfv41llcs8e2tll5" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how it projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. 

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to at least their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>fieaxmytdpznbcphfv41llcs8e2tll5</sha1>
    </revision>
    <revision>
      <id>2830676</id>
      <parentid>2830618</parentid>
      <timestamp>2026-09-03T05:14:39Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
      <origin>2830676</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="113731" sha1="jajq8w0isizhdp6ljtce0i73lhv23cg" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to at least their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Distribution of stars in our galaxy */</comment>
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      <text bytes="113722" sha1="puddqi86aqaanmdoiezjo0pwwz9toe3" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space.   
The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our sun and our galaxy) is contained in a smaller 3-sphere lying (we assume) near the largest 3-sphere's surface. We ourselves live within such a 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2830681</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="113770" sha1="mi8o3o1jlj1e7mj9ijo4p7k709run5y" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

The enclosing surface of a spherical region of 4-space of any size is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our galaxy and our sun) is contained in a smaller 3-sphere of its own lying (we assume) on the largest 3-sphere, forming a 4-dimensional lump in its 3-dimensional surface. We ourselves live within that expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-sphere manifold, as a 3-spherical shell within the moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2830684</id>
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        <username>Dc.samizdat</username>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="113954" sha1="g4w8prz86l7tw5yb6abdc1rib2pc1on" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects we observe (including our galaxy and our sun) is contained in a smaller 3-sphere of its own lying (we assume) on the largest 3-sphere, forming a 4-ball lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding as the largest 3-sphere inflates radially at velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our entire 3-manifold, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>g4w8prz86l7tw5yb6abdc1rib2pc1on</sha1>
    </revision>
    <revision>
      <id>2830688</id>
      <parentid>2830684</parentid>
      <timestamp>2026-09-03T06:35:13Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
      <origin>2830688</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="113958" sha1="6oxxvch9ddn6qroxh5f3tu3w9z4m41o" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller orbital 3-sphere of its own, lying (we assume) on the largest 3-sphere as a lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates radially at velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a tiny patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our solar system's 3-manifold, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>6oxxvch9ddn6qroxh5f3tu3w9z4m41o</sha1>
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    <revision>
      <id>2830691</id>
      <parentid>2830688</parentid>
      <timestamp>2026-09-03T06:41:01Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="113969" sha1="ja9ytf2z3xmwsqfki369kq796o2ruea" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our solar system's 3-manifold, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ja9ytf2z3xmwsqfki369kq796o2ruea</sha1>
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    <revision>
      <id>2830694</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="113972" sha1="5szmznr8rgrrrnf43st8b4rsztzhri1" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="113981" sha1="j6py1h2pvsk8ajv4rvfg07zb2xoeoqa" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="113981" sha1="lla1hepj4mimnfvbvtfjqgzc330u29h" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>lla1hepj4mimnfvbvtfjqgzc330u29h</sha1>
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    <revision>
      <id>2830704</id>
      <parentid>2830702</parentid>
      <timestamp>2026-09-03T07:23:00Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="113995" sha1="a8lv52zzdyhmer5bmweqf1q6by5gqjc" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. Consequently, for example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of a set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="113983" sha1="3qpfyhlvhk3w9h6bh6hoh8z1jl2xbgj" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of the set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="114148" sha1="lbos7zfl8kun4a29ejnb5yu98zktaku" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of the set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at velocity &lt;math&gt;c&lt;/math&gt; through their own proper time.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>lbos7zfl8kun4a29ejnb5yu98zktaku</sha1>
    </revision>
    <revision>
      <id>2830713</id>
      <parentid>2830711</parentid>
      <timestamp>2026-09-03T07:47:37Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Special relativity describes Euclidean 4-space */</comment>
      <origin>2830713</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="114335" sha1="mc1ojsuznhiislgweknnfe9min21qun" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of the set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity &lt;math&gt;c&lt;/math&gt; through their own proper time. Observed from another reference frame in relative motion, they appear to be moving in space and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2830714</id>
      <parentid>2830713</parentid>
      <timestamp>2026-09-03T07:49:46Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="114331" sha1="qgwy063n1ldkeqrqmd22utdy3y5rscp" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of the set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear to be moving in space and their clocks appear to be slowed to less than the maximum velocity &lt;math&gt;c&lt;/math&gt;.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="114328" sha1="354v0rodc6w7inrsou2897czrvwdnt6" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of the set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear to be moving in space, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="114358" sha1="37uomgm9yn8gqrqlq5csf4mvu2spw5o" xml:space="preserve">= Real Euclidean four-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt; =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - August 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as Euclidean space of four orthogonal spatial dimensions &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that physical space has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-displacements through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a 3-sphere expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a 4-ball. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere as a 4-dimensional lump embedded in its 3-dimensional surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch of a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. 

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from protons to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater our relative motion and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, stationary atoms exhibit the &lt;math&gt;SO(4)&lt;/math&gt; symmetries of discrete isoclinic (equi-angled) double rotations (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;) of the set of regular 4-polytopes characteristic of their [[w:Atomic_number|atomic number]].

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional &quot;space&quot; because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction. 

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper spacetime (3-space plus time) coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity &lt;math&gt;c&lt;/math&gt; relative to universal 4-coordinate space, so the maximum relative velocity between two observers is &lt;math&gt;2c&lt;/math&gt; when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to &lt;math&gt;c&lt;/math&gt;, it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions. In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. 

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time&lt;br&gt;
&lt;br&gt;
...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity&lt;br&gt;
&lt;br&gt;
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R^4}&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (intersect each other). When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are obvious analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.{{Efn|
...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it might be useful to do this here...}}

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be termed a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is also an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel planes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting planes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt;

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (the relative motion is a small fraction of the speed of light). These typical observations agree closely with the predictions of special relativity, because the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT). Since one of its two completely orthogonal rotations (Q) has such a long period, it is almost indistinguishable from a straight translation (T).{{Efn|
...this paragraph is probably misplaced here and should not interrupt the discussion at this point}}

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.{{Efn|
...cite Jesper Goransson's very concise paper}}

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions to turn in. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;math&gt;SO(3)&lt;/math&gt; and &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, in contrast to the &lt;math&gt;SO(4)&lt;/math&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;math&gt;\mathbb{R^3}&lt;/math&gt;, spherical space &lt;math&gt;S^3&lt;/math&gt; and Euclidean space &lt;math&gt;\mathbb{R^4}&lt;/math&gt;.

...

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity &lt;math&gt;c&lt;/math&gt;), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), breaking its stationary &lt;math&gt;SO(4)&lt;/math&gt; isoclinic rotation symmetry (&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;math&gt;SO(4)&lt;/math&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. But the Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;math&gt;\mathrm{T}&lt;/math&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;math&gt;SO(3)&lt;/math&gt; rotation component (&lt;math&gt;\mathrm{Q}&lt;/math&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a screw translation (&lt;math&gt;\mathrm{QT}&lt;/math&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;math&gt;\mathrm{T}&lt;/math&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt;) breaks to ... &lt;math&gt;S^3&lt;/math&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. 

That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion.{{Efn|

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;math&gt;SO(4)&lt;/math&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}}

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

Light is discrete reflections. Mass is discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Light is discrete reflections. Gravity and all forces are discrete rotations. Both are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. 

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. 

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, successive Euclidean spaces are dimensionally analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. 

This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may, or may not, all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a sense related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw displacements), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw displacement has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to ... &lt;math&gt;S^3&lt;/math&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;math&gt;H^3&lt;/math&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? ''What is the nature of the boundary which confines us to just three dimensions?'' 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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  <page>
    <title>Bully Metric Timestamps</title>
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      <timestamp>2026-09-02T12:51:35Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <text bytes="61606" sha1="sfx1kb2o0t0vntxhuksg7xof51cvl7h" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 1,554.5 parsecs of solar travel, which is traversed by the Sun in roughly 6.65 million years. The cyan lines at the bottom of the image further divide one sector into 16 equal subsectors of 97.156 parsecs each; this matches the 10&lt;sup&gt;10&lt;/sup&gt; light-second distance baseline previously established as typical for naked-eye stars.

The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction.

A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. To the right of the red line, nine cyan grid lines represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. To the left of the red line, five cyan grid lines represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000. These cyan grid lines are bounded on the right by a yellow line (representing timestamp 8200 0000 0000) and on the left by a yellow line representing timestamp 8210 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. The cyan lines at the bottom of the image further divide one sector into 10 equal subsectors of 100 parsecs each; this is roughly the baseline distance typical for naked-eye stars.

The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction.

A red grid line represents the polar angle the Sun would have had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. To the left of the red line, nine cyan lines respectively represent the polar angle of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>sfx1kb2o0t0vntxhuksg7xof51cvl7h</sha1>
    </revision>
    <revision>
      <id>2830538</id>
      <parentid>2830489</parentid>
      <timestamp>2026-09-02T16:47:46Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* The Milky Way */</comment>
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      <text bytes="61695" sha1="3gc0d37cqqsjtcqnc8gt8lpmnl9d0u3" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.


&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. The cyan lines at the bottom of the image further divide one sector into 10 equal subsectors of 100 parsecs each; this is roughly the baseline distance typical for naked-eye stars.

The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction.

A red grid line represents the polar angle the Sun would have had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. To the left of the red line, nine cyan lines respectively represent the polar angle of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>3gc0d37cqqsjtcqnc8gt8lpmnl9d0u3</sha1>
    </revision>
    <revision>
      <id>2830546</id>
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      <timestamp>2026-09-02T18:32:26Z</timestamp>
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        <username>Unitfreak</username>
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      <comment>/* The Milky Way */</comment>
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      <text bytes="61694" sha1="26j73czqy7nz7r1od105kxexm365bn7" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. The cyan lines at the bottom of the image further divide one sector into 10 equal subsectors of 100 parsecs each; this is roughly the baseline distance typical for naked-eye stars.

The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction.

A red grid line represents the polar angle the Sun would have had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. To the left of the red line, nine cyan lines respectively represent the polar angle of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>26j73czqy7nz7r1od105kxexm365bn7</sha1>
    </revision>
    <revision>
      <id>2830547</id>
      <parentid>2830546</parentid>
      <timestamp>2026-09-02T18:48:39Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Galactic Weeks */</comment>
      <origin>2830547</origin>
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      <text bytes="61888" sha1="f1ngtp6c1ilcs1p28bmg3979t9tkazo" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>f1ngtp6c1ilcs1p28bmg3979t9tkazo</sha1>
    </revision>
    <revision>
      <id>2830557</id>
      <parentid>2830547</parentid>
      <timestamp>2026-09-02T19:22:43Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62103" sha1="o6vp27un6awfgt6ek5l2tu96m4l5tur" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>o6vp27un6awfgt6ek5l2tu96m4l5tur</sha1>
    </revision>
    <revision>
      <id>2830559</id>
      <parentid>2830557</parentid>
      <timestamp>2026-09-02T19:26:03Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62103" sha1="7igwmbtgg9ceoyzufjyxnrs1uk3k8a1" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 10px auto 10px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
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      <text bytes="62120" sha1="82ez11ivndxjji23touh2offp4tnp1d" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; cellpadding=&quot;10&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>82ez11ivndxjji23touh2offp4tnp1d</sha1>
    </revision>
    <revision>
      <id>2830561</id>
      <parentid>2830560</parentid>
      <timestamp>2026-09-02T19:31:35Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62103" sha1="o6vp27un6awfgt6ek5l2tu96m4l5tur" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
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      <id>2830563</id>
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        <username>Unitfreak</username>
        <id>695864</id>
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      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62101" sha1="q81t6n5asoa6l7yi8llvqc1cgltyibg" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 5px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 5px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 10px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>q81t6n5asoa6l7yi8llvqc1cgltyibg</sha1>
    </revision>
    <revision>
      <id>2830564</id>
      <parentid>2830563</parentid>
      <timestamp>2026-09-02T19:35:29Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62099" sha1="nkr2u54oipyhaw4fsfgctbhb9k3p8bm" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in  parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>nkr2u54oipyhaw4fsfgctbhb9k3p8bm</sha1>
    </revision>
    <revision>
      <id>2830569</id>
      <parentid>2830564</parentid>
      <timestamp>2026-09-02T19:56:25Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
      <origin>2830569</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="62076" sha1="th0tc6uurj5lquvxcylw9qqwilbacjx" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 1 &lt;math&gt;R_{\odot}&lt;/math&gt; per Bully timestamp&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>th0tc6uurj5lquvxcylw9qqwilbacjx</sha1>
    </revision>
    <revision>
      <id>2830571</id>
      <parentid>2830569</parentid>
      <timestamp>2026-09-02T19:57:15Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
      <origin>2830571</origin>
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      <text bytes="62066" sha1="aol2bqx9bpn164rcg73cw6cs4eivcsv" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 1 solar radius per Bully timestamp&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>aol2bqx9bpn164rcg73cw6cs4eivcsv</sha1>
    </revision>
    <revision>
      <id>2830576</id>
      <parentid>2830571</parentid>
      <timestamp>2026-09-02T20:11:11Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
      <origin>2830576</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="62328" sha1="klnzsnxr71svqeki9i7sp420gc5zswr" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 1 solar radius per Bully timestamp&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.597453&lt;/sup&gt; ≈ 396,635 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.597453&lt;/sup&gt; ≈ 24,789.7 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.597453&lt;/sup&gt; ≈ 1,549.36 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.597453&lt;/sup&gt; ≈ 49,579 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>klnzsnxr71svqeki9i7sp420gc5zswr</sha1>
    </revision>
    <revision>
      <id>2830577</id>
      <parentid>2830576</parentid>
      <timestamp>2026-09-02T20:12:27Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62298" sha1="ekpnyh00fdtamxa86u20plfthwabkbs" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 1 solar radius per Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 1 solar radius per Bully timestamp&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.597453&lt;/sup&gt; ≈ 396,635 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.597453&lt;/sup&gt; ≈ 24,789.7 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.597453&lt;/sup&gt; ≈ 1,549.36 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.597453&lt;/sup&gt; ≈ 49,579 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | &lt;small&gt;'''2&lt;sup&gt;25.333776&lt;/sup&gt;''' ≈ 42,288,909&lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;-0.063992&lt;/sup&gt; ≈ 0.956613 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0&lt;/sup&gt; = 1 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
      <sha1>ekpnyh00fdtamxa86u20plfthwabkbs</sha1>
    </revision>
    <revision>
      <id>2830591</id>
      <parentid>2830577</parentid>
      <timestamp>2026-09-02T20:31:42Z</timestamp>
      <contributor>
        <username>Unitfreak</username>
        <id>695864</id>
      </contributor>
      <comment>/* Is the Bully system internally consistent? */</comment>
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      <text bytes="62275" sha1="i9vhmdp4881l55irwdcp0859vcsw19a" xml:space="preserve">&lt;small&gt;[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt; &lt;/small&gt;

The '''Bully Metric Timestamp''' system is an alternative timekeeping framework that utilizes the orbit of the Sun around the Milky Way Galaxy to mark the passage of time. A new successive Bully timestamp is realized each time the Sun advances by roughly one solar radius along its path through the Cosmos. Using '''12-digit''' [[w:hexadecimal|hexadecimal]] timestamps, the Bully system has enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is:
&amp;thinsp;
:&lt;math&gt;16^{12} \times 3,055 \text{ sec} \approx 27.25 \text{ billion years}&lt;/math&gt;


[[File:Bully_Metric_Galactic_Orbit_1_Timestamp.png|thumb|center|450px|alt=Diagram showing the Sun advancing a distance equal to its own radius along its galactic trajectory over a period of 3055 seconds.|'''Figure 1:''' Motion of the Sun between two successive Bully timestamps.]]

== One Solar Radius ==
The Sun hurtles around the Milky Way galaxy at a blistering 0.076% the speed of light (227.7 kilometers per second). And yet, even at that staggering pace, it takes about five-sixths of an hour (3,055 seconds) for the Sun to cross a distance equal to its own radius. This highlights the truly colossal size of our star, which boasts a radius of 2.3206 light-seconds (695,700 kilometers).

'''Figure 1''' illustrates the physical movement of the Sun between two successive Bully timestamps. Timestamp '''8209 ED00 0000''' is defined to have occurred at exactly '''12:00:00 TAI (International Atomic Time) on June 21, 1998'''. The sequential timestamp, '''8209 ED00 0001''', occurred exactly 3,055 seconds later at '''12:50:55 TAI on June 21, 1998'''. The Sun orbits a distance of roughly one solar radius during each 3,055-second period.

The following mnemonics are aids to help one remember the length of the solar radius (&lt;math&gt;R_{\odot}&lt;/math&gt;) in [[W:astronomical units|astronomical units ]] (AU) and in [[W:light-second|light-seconds]] (ls); the time required for the Sun to travel a distance equal to its own radius (&lt;math&gt;\tau_{R\odot}&lt;/math&gt;); and the orbital velocity (&lt;math&gt;V_\odot&lt;/math&gt;) relative to the speed of light (c):

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \approx \frac{499}{215} \text{ ls} \approx 2.3209 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
1 \, R_{\odot} \sim \frac{10^{10}}{16^{8}} \text{ ls} \approx 2.33 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;\tau_{R\odot} \equiv 3055 \text{ seconds}&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
V_{\odot} \equiv \frac{R_{\odot}}{\tau_{R\odot}} \approx 0.07597\% \, c
&lt;/math&gt;
&amp;hairsp;

=== The Heliosphere ===
The heliosphere, it turns out, is not actually a sphere. As shown in '''Figure 2''', it is a vast, oblong, tailed, bubble-like region extending from the Sun into the surrounding space. The heliosphere is somewhat analogous to Earth's atmosphere; however, Earth's atmosphere is a comparatively thin layer of gas that remains near the planetary surface, whereas the heliosphere is a plasma constantly blasted out into space by the extreme heat and energy of the Sun.

[[File:Bully_Metric_Galactic_Orbit_65536_Timestamps.png|thumb|right|450px|alt=Diagram showing the Sun traveling through the oblong shape of the heliosphere over a span of 16 to the 4th power timestamps.|'''Figure 2:''' Motion of the Sun during the passage of 16&lt;sup&gt;4&lt;/sup&gt; Bully timestamps.]]

The heliosphere is so vast that if it were truly spherical, its diameter would be on an order of magnitude similar to '''16&lt;sup&gt;4&lt;/sup&gt; solar radii'''. The digit in the '''fifth position''' in a Bully timestamp represents the time required for the Sun to orbit for '''6.344 years''', which covers a distance of approximately one spherical heliosphere.

Figure 2 illustrates the orbit of the Sun (Sun not drawn to scale) over a period of 6.344 years. As explained previously, timestamp 8209 ED00 0000 is defined to have occurred at exactly 12:00:00 TAI on June 21, 1998. Timestamp '''8209 ED01 0000''' therefore occurred roughly 6.344 years later at '''18:34:40 TAI on October 24, 2004'''.

Incidentally, the Voyager 1 spacecraft crossed into the heliosheath, as shown in Figure 2, on December 16th 2004. Both Voyager spacecraft (Voyager 1 and Voyager 2) have since crossed entirely out of the heliosphere and entered the surrounding interstellar space.

The following mnemonics aid in remembering the size of the heliosphere and the time duration required for the Sun to orbit that distance:

&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \approx \frac{16^{4}}{215} \text{ AU} \approx 304.82 \text{ AU}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
16^{4}\,R_{\odot} \sim \frac{10^{10}}{16^{4}} \text{ ls} \approx 152,000 \text{ ls}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{4}\,\tau_{R\odot} &amp;\equiv 16^{4} \times 3,055 \text{ s} \\
&amp;\approx 6.344 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

=== Naked-Eye Stars ===
'''Figure 3a''' illustrates the physical movement of the Sun (Sun not drawn to scale) between 16&lt;sup&gt;8&lt;/sup&gt; successive Bully timestamps. It is estimated that timestamp '''8209 0000 0000''' would have occurred approximately 383,000 B.C., and timestamp '''820A 0000 0000''' is estimated to occur around 33,000 A.D., for a total time lapse of '''416,000 years'''. The stacked histogram in Figure 3a has a cyan dashed line marking 97.156 parsecs, which is 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, or roughly the distance the Sun travels in 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. A parsec is a common length unit used in astronomy and is defined such that &lt;math&gt; 20 \pi \text{ parsecs} = 60^{4} \text{ AU} &lt;/math&gt;.

As indicated in the histogram, a large percentage of naked-eye stars are closer to the Sun than 97.156 parsecs, meaning the appearance of the night sky completely changes over this timeframe.

[[File:Bully_Metric_Galactic_Orbit_4294967296_Timestamps.png|thumb|center|600px|alt=Diagram showing a stacked histogram of naked-eye stars binned according to brightness and distance from the sun. A large percentage of these stars are closer to the sun than 10&lt;sup&gt;10&lt;/sup&gt; light-seconds, which is the distance the sun travels in 16^8 Bully timestamps.|'''Figure 3a:''' Motion of the Sun during the passage of 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps. The included stacked histogram shows that a large percentage of naked-eye stars are within this travel distance of the sun, 97.156 parsecs or 10&lt;sup&gt;10&lt;/sup&gt; light-seconds.]]

The following mnemonics indicate a typical distance of naked-eye stars and the time required for the Sun to travel that distance:

&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,R_{\odot} &amp;\sim 10^{10} \text{ ls} \\
&amp;\approx 97.156 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
16^{8}\,\tau_{R\odot} &amp;\equiv 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 415,792 \text{ years}
\end{align}
&lt;/math&gt;
&amp;hairsp;

[[Bully_Metric_Naked-Eye_Stars|Learn More About the Meaning of Naked-Eye Stars]]

==== The Milky Way ====
[[File:Milky_Way_map_by_Gaia_High_Density_Grid.jpg|thumb|center|600px|'''Figure 3e''': Division of the Milky Way into 32 equal polar sectors (yellow grid).]]

'''Figure 3e''' illustrates the division of the Milky Way into 32 equal polar sectors (yellow grid). Each sector represents about 16 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds of solar travel, which is traversed by the Sun in roughly 16&lt;sup&gt;9&lt;/sup&gt; Bully timestamps (6.65 million years). Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector of the yellow grid into 16 equal subsectors of 10&lt;sup&gt;10&lt;/sup&gt; light-seconds each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3e. The Sun orbits in a clockwise direction. A red grid line represents the polar angle the Sun will have during Bully timestamp 820A 0000 0000, which is estimated to occur around 33,000 A.D. The nine cyan grid lines to the right of the red line represent the Sun's polar angle during Bully timestamps 8201 0000 0000 through 8209 0000 0000. The five cyan grid lines to the left of the red line represent the Sun's polar angle during Bully timestamps 820B 0000 0000 through 820F 0000 0000.

The Sun's full orbital path around the Milky Way is roughly 49,744 parsecs and is completed in roughly 212.8 million years.

&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,R_{\odot} &amp;\sim {512 \times 10^{10}} \text{ ls} \\
&amp;\approx 49,744 \text{ parsecs}
\end{align}
&lt;/math&gt;
&amp;hairsp;
:&lt;math&gt;
\begin{align}
512 \times 16^{8}\,\tau_{R\odot} &amp;\equiv 512 \times 16^{8} \times 3,055 \text{ s} \\
&amp;\approx 212.8 \text{ million years} \\
\end{align}
&lt;/math&gt;
&amp;hairsp;

== The Galactic Calendar ==
[[File:Galactic centre orbits.svg|thumb|300px|'''Figure 4a''':Inferred orbits of 6 stars around the supermassive black hole Sagittarius A* at the Milky Way's center&lt;ref name=&quot;Eisenhauer&quot;&gt;{{cite journal|last=Eisenhauer|first=F.|display-authors=et al. |title=SINFONI in the Galactic Center: Young Stars and Infrared Flares in the Central Light-Month|journal=The Astrophysical Journal|date=July 20, 2005|volume=628|issue=1|pages=246–259|doi=10.1086/430667|bibcode=2005ApJ...628..246E|arxiv=astro-ph/0502129|s2cid=122485461 }}&lt;/ref&gt;]]
[[W:Andrea Ghez|Andrea Ghez]] and [[W:Reinhard Genzel|Reinhard Genzel]] were jointly awarded one-half of the 2020 Nobel Prize in Physics for their discovery of a supermassive compact object—now universally known as the black hole Sagittarius A*—at the center of the Milky Way. Using the world’s largest telescopes, both teams observed stars whipping around an invisible, incredibly heavy mass at extreme speeds. One star, labeled S2 in '''Figure 4a''', completes an orbit in just 16 years. Their calculations revealed that an entity of roughly 4 million solar masses is packed into an area no larger than our solar system, offering definitive evidence of a supermassive black hole.

By meticulously mapping the complete 3D elliptical orbits of stars over decades, Reinhard Genzel's research team calculated the exact distance from the Sun to the Galactic Center to an unprecedented degree of accuracy. They narrowed the measurement down to [https://www.mpe.mpg.de/6588951/The-black-hole 8,275 parsecs] (or 26,990 light-years), with a margin of error of less than 1%. If the Sun were assumed to follow a perfectly circular orbit around Sagittarius A*, with a constant radial distance of 8,275 parsecs, the total circumference of that ideal orbit would be determined by multiplying the radius by 2π:
&amp;hairsp;
:&lt;math&gt;\begin{align}
{\text{Circumference}} &amp;= 2\pi \times 8,275{\text{ parsecs}} \\
&amp;\approx 51,993{\text{ parsecs}} \\
\end{align}&lt;/math&gt;
&amp;hairsp;
If we divide this '''roughly 52,000-parsec''' idealized orbit into &quot;Galactic Weeks&quot;, where each week represents 1,000 parsecs of orbital travel, then a full Galactic Year would consist of 52 weeks. This beautifully mirrors the structure of an Earth year, which is also composed of roughly 52 weeks.

==== Galactic Weeks ====
[[File:Milky_Way_map_by_Gaia_High_Density_Weeks_Grid.jpg|thumb|center|600px|'''Figure 3f''': Division of the Milky Way into 52 equal galactic weeks of solar travel (yellow grid).]]

'''Figure 3f''' illustrates the division of the Milky Way into 52 equal galactic weeks (yellow grid). Each week represents 1,000 parsecs of solar travel, which is traversed by the Sun in roughly 4.09 million years. Three of the grid lines are labeled as Bully timestamps '''8200 0000 0000''', '''8280 0000 0000''', and '''8300 0000 0000''', respectively indicating the estimated polar angle of the Sun at approximately 4.1 million years B.C., 49 million years A.D., and 102 million years A.D.

The cyan lines in the lower half of the image further divide one sector into 10 equal subsectors of 100 parsecs each. The Sun is currently located directly below the Galactic Center, at the 6 o'clock position in Figure 3f, and orbits in a clockwise direction. A red grid line represents the polar angle the Sun had during Bully timestamp 8209 D89D 89D8 (approximately 31,000 B.C.), which is 1,000 parsecs of travel beyond the yellow line representing timestamp 8200 0000 0000. The nine cyan lines to the left of the red line represent the polar angles of the Sun after traveling 1,100 through 1,900 parsecs beyond timestamp 8200 0000 0000.

=== Idealized Galactic Years ===
Within the context of Bully timekeeping, an idealized '''Bully Galactic Year''' is defined to have a duration of exactly '''2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps''' (approximately 213 million years), and the Sun is assumed to follow an orbital path of exactly 52,000 parsecs. The table in '''Figure 4b''' illustrates the division of an idealized Galactic Year into 52 equal portions. The table shows the Bully timestamp at which each 1,000 parsecs of travel distance would be achieved in this idealized orbit.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; width:100%; max-width:800px;&quot;
|+ '''Figure 4b:''' The 66th Bully Galactic Calendar

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|1st Quarter}} || {{nowrap|2nd Quarter}} || {{nowrap|3rd Quarter}} || {{nowrap|4th Quarter}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 0}} || {{color|blue|''0 parsecs''}} &lt;br/&gt;'''{{nowrap|8200 0000 0000}}''' || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8280 0000 0000}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8300 0000 0000}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8380 0000 0000}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 1}} || {{color|blue|''1000 parsecs''}} &lt;br/&gt; '''{{nowrap|8209 D89D 89D8}}''' || {{color|blue|''14,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8289 D89D 89D8}}''' || {{color|blue|''27,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8309 D89D 89D8}}''' || {{color|blue|''40,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8389 D89D 89D8}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 2}} || {{color|blue|''2,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8213 B13B 13B1}}''' || {{color|blue|''15,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8293 B13B 13B1}}''' || {{color|blue|''28,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8313 B13B 13B1}}''' || {{color|blue|''41,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8393 B13B 13B1}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 3}} || {{color|blue|''3,000 parsecs''}} &lt;br/&gt; '''{{nowrap|821D 89D8 9D89}}''' || {{color|blue|''16,000 parsecs''}} &lt;br/&gt; '''{{nowrap|829D 89D8 9D89}}''' || {{color|blue|''29,000 parsecs''}} &lt;br/&gt; '''{{nowrap|831D 89D8 9D89}}''' || {{color|blue|''42,000 parsecs''}} &lt;br/&gt; '''{{nowrap|839D 89D8 9D89}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 4}} || {{color|blue|''4,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8227 6276 2762}}''' || {{color|blue|''17,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82A7 6276 2762}}''' || {{color|blue|''30,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8327 6276 2762}}''' || {{color|blue|''43,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83A7 6276 2762}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 5}} || {{color|blue|''5,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8231 3B13 B13B}}''' || {{color|blue|''18,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82B1 3B13 B13B}}''' || {{color|blue|''31,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8331 3B13 B13B}}''' || {{color|blue|''44,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83B1 3B13 B13B}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 6}} || {{color|blue|''6,000 parsecs''}} &lt;br/&gt; '''{{nowrap|823B 13B1 3B13}}''' || {{color|blue|''19,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82BB 13B1 3B13}}''' || {{color|blue|''32,000 parsecs''}} &lt;br/&gt; '''{{nowrap|833B 13B1 3B13}}''' || {{color|blue|''45,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83BB 13B1 3B13}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 7}} || {{color|blue|''7,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8244 EC4E C4EC}}''' || {{color|blue|''20,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82C4 EC4E C4EC}}''' || {{color|blue|''33,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8344 EC4E C4EC}}''' || {{color|blue|''46,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83C4 EC4E C4EC}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 8}} || {{color|blue|''8,000 parsecs''}} &lt;br/&gt; '''{{nowrap|824E C4EC 4EC4}}''' || {{color|blue|''21,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82CE C4EC 4EC4}}''' || {{color|blue|''34,000 parsecs''}} &lt;br/&gt; '''{{nowrap|834E C4EC 4EC4}}''' || {{color|blue|''47,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83CE C4EC 4EC4}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 9}} || {{color|blue|''9,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8258 9D89 D89D}}''' || {{color|blue|''22,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82D8 9D89 D89D}}''' || {{color|blue|''35,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8358 9D89 D89D}}''' || {{color|blue|''48,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83D8 9D89 D89D}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 10}} || {{color|blue|''10,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8262 7627 6276}}''' || {{color|blue|''23,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82E2 7627 6276}}''' || {{color|blue|''36,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8362 7627 6276}}''' || {{color|blue|''49,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83E2 7627 6276}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 11}} || {{color|blue|''11,000 parsecs''}} &lt;br/&gt; '''{{nowrap|826C 4EC4 EC4E}}''' || {{color|blue|''24,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82EC 4EC4 EC4E}}''' || {{color|blue|''37,000 parsecs''}} &lt;br/&gt; '''{{nowrap|836C 4EC4 EC4E}}''' || {{color|blue|''50,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83EC 4EC4 EC4E}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|Week 12}} || {{color|blue|''12,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8276 2762 7627}}''' || {{color|blue|''25,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82F6 2762 7627}}''' || {{color|blue|''38,000 parsecs''}} &lt;br/&gt; '''{{nowrap|8376 2762 7627}}''' || {{color|blue|''51,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83F6 2762 7627}}'''
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | End of Quarter || {{color|blue|''13,000 parsecs''}} &lt;br/&gt; '''{{nowrap|827F FFFF FFFF}}''' || {{color|blue|''26,000 parsecs''}} &lt;br/&gt; '''{{nowrap|82FF FFFF FFFF}}''' || {{color|blue|''39,000 parsecs''}} &lt;br/&gt; '''{{nowrap|837F FFFF FFFF}}''' || {{color|blue|''52,000 parsecs''}} &lt;br/&gt; '''{{nowrap|83FF FFFF FFFF}}'''
|}

Timestamps in the range '''8200 0000 0000''' through '''83FF FFFF FFFF''' indicate that the system is recording time within the '''66th Bully Galactic Year''' of the Universe. However, the Sun (and our solar system) did not come into existence until the 45th Bully Galactic Year, meaning our solar system is only '''21 Bully Galactic Years old'''.

==== Is the Galactic Calendar Realistic? ====
[[File:Sun_in_orbit_around_Galactic_Centre.gif|thumb|right|300px|alt=Diagram showing multiple stars moving along their respective orbital paths around the Galactic Center over a span of 250 million years.|'''Figure 4c:''' Stars orbiting around the Galactic Center during a 250 million-year time period.]]

The duration of time required for the Sun (or any other star) to orbit once around the center of the Milky Way Galaxy is not a fixed constant, but rather depends on the path a particular star follows as it orbits (see Figure 4c). Stars closer to the center orbit much more quickly than those on the outer edges. The stars shown in '''Figure 4c''' all eventually localize near the Sun despite having vastly different orbital trajectories, visually illustrating the long-term subtlety of galactic orbits.

Because the Sun’s deep-time trajectory is slightly chaotic and unpredictable, an operational offset will always exist between the passage of Bully time and physical observations of the Sun's galactic displacement. Therefore, while the table in Figure 4b states that the galactic week beginning on timestamp '''{{nowrap|8209 D89D 89D8}}''' corresponds to 1,000 parsecs of displacement, this relationship must be understood as an estimate.

In practice, even if the system were calibrated so that timestamp '''{{nowrap|8209 D89D 89D8}}''' perfectly aligned with the exact moment the Sun traveled 1,000 parsecs, this precise alignment would immediately begin to decay. The subsequent milestone at timestamp '''{{nowrap|8213 B13B 13B1}}''' would almost certainly not occur at the exact instant the Sun reached the 2,000-parsec mark.

==== Is the Bully system internally consistent? ====
In Figure 3e, the Sun is shown to travel roughly 512 × 10&lt;sup&gt;10&lt;/sup&gt; light-seconds (49,744 parsecs) per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps (213 million years). However, the Bully Galactic Calendar shown in Figure 3f assumes exactly 52,000 parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; timestamps, introducing a distinct discrepancy in both galactic scale and calculated orbital velocity.

Because the long-term orbital dynamics of the Sun are subject to complex gravitational perturbations, standard stellar movement is neither perfectly uniform nor entirely predictable. Consequently, the Sun's true orbital velocity remains a subject of ongoing discovery and refinement. The conjectured values used in Figure 3e and Figure 3f should be viewed as '''practical assumptions''' rather than a reflection of a stable, long-term physical reality; the idealized Bully Calendar is a '''conceptual model''' designed to help visualize the immense scale of the galactic orbit.

The table in '''Figure 4d''' illustrates how scaling the assumed baseline velocity from 1 solar radius per Bully timestamps up to 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps aligns the highest digits with rounded integer multiples of the parsec length.

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; border-collapse: collapse; font-family: sans-serif;&quot;

|+ style=&quot;font-weight: bold; margin-bottom: 8px;&quot; | '''Figure 4d''': Distance Conversions to Parsecs

! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt;Bully Timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 1 solar radius per Bully timestamp&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 10&lt;sup&gt;10&lt;/sup&gt; light-seconds per 16&lt;sup&gt;8&lt;/sup&gt; Bully timestamps&lt;/small&gt;
! style=&quot;background-color: #f2f2f2;{{Text color default}}; text-align: left; padding: 3px;&quot; | &lt;small&gt; Orbital distance in parsecs assuming 52,000 parsecs per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps&lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;11&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;18.597453&lt;/sup&gt; ≈ 396,635 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.602232&lt;/sup&gt; ≈ 397,951 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;18.666224&lt;/sup&gt; ≈ 416,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;10&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;14.597453&lt;/sup&gt; ≈ 24,789.7 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.602232&lt;/sup&gt; ≈ 24,871.9 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;14.666224&lt;/sup&gt; ≈ 26,000.0 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''16&lt;sup&gt;9&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;10.597453&lt;/sup&gt; ≈ 1,549.36 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.602232&lt;/sup&gt; ≈ 1,554.50 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;10.666224&lt;/sup&gt; ≈ 1,625.00 &lt;/small&gt;

|- style=&quot;background-color: #e6f2ff;{{Text color default}}; font-weight: bold;&quot;
! colspan=&quot;4&quot; style=&quot;text-align: left; padding: 8px;&quot; | Off Nominal Values

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;41&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;15.597453&lt;/sup&gt; ≈ 49,579 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.602232&lt;/sup&gt; ≈ 49,744 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;15.666224&lt;/sup&gt; ≈ 52,000 &lt;/small&gt;

|-
| style=&quot;text-align: left; padding: 8px;&quot; | '''2&lt;sup&gt;26&lt;/sup&gt;'''
| &lt;small&gt;2&lt;sup&gt;0.597453&lt;/sup&gt; ≈ 1.513043 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0.602232&lt;/sup&gt; ≈ 1.518064 &lt;/small&gt;
| &lt;small&gt;2&lt;sup&gt;0.666224&lt;/sup&gt; ≈ 1.586914 &lt;/small&gt;
|}

== Anchoring Bully Timestamps ==
To establish a rigid temporal framework, the Bully system is anchored by selecting timestamp '''{{nowrap|8209 ED00 0000}}''' to coincide precisely with '''12:00:00 TAI on June 21, 1998'''. Following this initial anchoring, the progression of all subsequent Bully timestamps is maintained uniformly via terrestrial atomic clocks, advancing by exactly one unit every '''3,055 TAI seconds'''. The following subsections will explain why timestamp '''{{nowrap|8209 ED00 0000}}''' was selected and anchored near the '''June solstice in 1998'''.

=== The Galactic Ecliptic Node near Sagittarius ===
'''Figure 5a''' depicts the 6.98-degree angular separation that exists between Sagittarius A* (the supermassive black hole at the center of the Milky Way) and the descending node of our Solar System’s [[W:Invariable_plane|Laplace invariable plane]] where it intersects the Galactic equator.

[[File:Sagittarius_A*_and_adjacent_Galactic_Ecliptic_Node.png|thumb|center|600px|alt=An educational image illustrating the 6.98-degree separation between Sagittarius A* and the adjacent Galactic Ecliptic Node. The Node, moving in concert with the Sun, shifts away from Sagittarius A* at a rate of 2.70 mas per year in right ascension and 5.60 mas per year in declination.|'''Figure 5a:''' A diagram showing the 6.98-degree angular separation between Sagittarius A* and the descending node of the Solar System's Laplace invariable plane.]]

As the Sun orbits the Galactic Center, the Galactic Ecliptic Node of the Solar System's invariable plane—moving in concert with the Sun—shifts away from Sagittarius A* at a rate of 2.70 mas (milliarcseconds) per year in right ascension and 5.60 mas per year in declination. From the perspective of the Sun, the node appears to be stationary, and the supermassive black hole appears to be moving in the opposite direction. In reality, it is the Solar System and the node that are moving.

==== A surrogate for the Sun ====
The path of the Solar System's Galactic Ecliptic Node as it shifts away from Sagittarius A* can be used as a surrogate to track the motion of the Sun. The node is currently located 6.9803° away from Sagittarius A*. The Sun's orbital travel distance is calculated by multiplying 6.9803° by the orbital radius (8,275 parsecs) and the ratio of radians to degrees (2π / 360°):

&lt;math&gt;
\begin{aligned}
d &amp;= 8,275 \text{ pc} \times 6.9803^\circ \times \left(\frac{2\pi}{360^\circ}\right) \\
&amp;\approx 1,008.14 \text{ pc}
\end{aligned}
&lt;/math&gt;

Based on this calculation, the Galactic Ecliptic Node—and by extension, the Sun—has traveled 1,008.14 parsecs in its orbit around the Galactic Center. According to the reference table in '''Figure 4b''', this 1,008.14-parsec distance falls beyond the 1,000-parsec milestone associated with timestamp '''{{nowrap|8209 D89D 89D8}}''', indicating that we have completed the zeroth week of the 66th Bully Galactic Year.

To pinpoint a more exact location, the table in '''Figure 5b''' provides a finer-grained increment. Our Sun's 1,008.14-parsec travel distance is larger than the '''1,007.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 ECC7 C23E}}''', but it is smaller than the '''1,008.87 parsecs''' corresponding to timestamp '''{{nowrap|8209 EF4D 094B}}'''. (Note: Figure 4a assumes an idealized travel distance of exactly 52,000 (2&lt;sup&gt;15.666224&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps, whereas Figure 5b uses the calculated distance of 51,993 (2&lt;sup&gt;15.666040&lt;/sup&gt;) parsecs of orbital travel per 2&lt;sup&gt;41&lt;/sup&gt; Bully timestamps.)

{| class=&quot;wikitable&quot; style=&quot;margin: 20px auto 40px auto; text-align:center; max-width:300px;&quot;
|+ '''Figure 5b:''' Week one, 66th Galactic Year

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Year 66 &lt;br /&gt; Galactic
|| {{nowrap|Bully timestamp}} || Solar Distance Traveled
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|One Week}} ||'''{{nowrap|8209 D89D 89D8}}''' || {{nowrap|{{color|blue|''999.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.007 Weeks}} ||'''{{nowrap|8209 EA42 7B32}}''' || {{nowrap|{{color|blue|''1006.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.008 Weeks}} ||'''{{nowrap|8209 ECC7 C23E}}''' || {{nowrap|{{color|blue|''1007.87 parsecs''}}}} 
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.009 Weeks}} ||'''{{nowrap|8209 EF4D 094B}}''' || {{nowrap|{{color|blue|''1008.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.010 Weeks}} ||'''{{nowrap|8209 F1D2 5058}}''' || {{nowrap|{{color|blue|''1009.87 parsecs''}}}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|1.100 Weeks}} ||'''{{nowrap|820A D4AD 4AD4}}''' || {{nowrap|{{color|blue|''1099.86 parsecs''}}}}
|}

{{Quote box| align  = center| width  = full| title  = Bully Timestamp Anchor| quote  = Bully timestamp '''{{nowrap|8209 ED00 0000}}''' was selected to be the timestamp anchor of the entire Bully system because it closely aligns with the 1,008.14-parsec orbital travel distance of the Sun (see table in Figure 5b).}}

=== Bullies in the Bully System ===
A planetary system's Laplace invariable plane passes through its barycenter (center of mass) and is strictly perpendicular to its total angular momentum vector. In our Solar System, the four giant planets account for 98% of this total angular momentum: Jupiter contributes the bulk at 60.3%, followed by Saturn (24.5%), Neptune (7.9%), and Uranus (5.3%). The individual descending node of each giant planet's ecliptic where it intersects the Galactic Equator is shown in Figure 5a:

* '''Invariable Plane Node (+)''': Marked with a large plus sign.
* '''Jupiter (♃)''': Positioned slightly to the right of the invariable plane's node.
* '''Uranus (⛢)''': Positioned to the right of Jupiter.
* '''Saturn (♄)''': Positioned on the inner left.
* '''Neptune (♆)''': Positioned on the far left.

As noted in the Merriam-Webster dictionary, the word &quot;bully&quot; had a positive connotation through much of history:
{{Blockquote|text=The earliest meaning of English bully was &quot;sweetheart&quot;. The word was probably borrowed from Dutch boel, &quot;lover&quot;. Later bully was used for anyone who seemed a good fellow, then for a blustering daredevil. Today, a bully is usually one whose claims to strength and courage are based on the intimidation of those who are weaker. “Bully.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/bully. Accessed 19 Aug. 2026.}}

Large astronomical objects—such as Sagittarius A*, the Sun, and giant planets like Jupiter and Saturn—can be thought of as bullies, both in the historical meaning of &quot;excellent&quot; or &quot;beautiful&quot; and in the modern meaning of being intimidating and threatening. 

{{Quote box| align  = center | width  = full | title  = The &quot;Bully&quot; Name | quote  = The specific &quot;bullies&quot; in the &quot;Bully&quot; timestamp system are [[w:Sagittarius A*|Sagittarius A*]], the [[w:Sun|Sun]], and the Solar System's [[w:Giant planet|giant planets]].}}

=== The Earth and Moon ===
[[File:Astronomical_unit_svg.svg|thumb|right|400px|'''TBD:''' The grey line indicates the Earth–Sun distance, which on average is about 1&amp;nbsp;astronomical unit.]]
The motions of the Earth and Moon are not suitable for precise, long-term time measurement due to deep time gravitational interactions. For example, tidal friction gradually slows the Earth's rotation and causes the Moon to drift farther away, making legacy day and month units unstable over millions of years. While the Bully timestamp system is not directly anchored to the motions of the Earth and Moon, it was developed with these motions in mind and incorporates a few of their unique characteristics.

&amp;hairsp;
:&lt;math&gt;
1\text{ AU} \approx 499 \text{ ls} \approx 215 \, R_{\odot} 
&lt;/math&gt;
&amp;hairsp;

==== Earth's sidereal year ====
The duration of Earth's sidereal year is '''31,558,149.76 seconds'''. While gravitational perturbations from neighboring planets cause this value to vary by 20 to 25 minutes annually, the averaged century-over-century lengthening is a mere 9.6 milliseconds. Given this relative stability, using a clean divisor of the sidereal year as the fundamental unit of the Bully timestamp system offers significant utility. Specifically, 3,055 seconds is an exact divisor of 31,558,150 seconds, meaning Earth's sidereal year—rounded to the nearest second—equals '''exactly 10,330 Bully timestamps'''.

{{Quote box| align  = center| width  = 100%| title  = Bully Timestamp Duration
| quote  = {{ordered list
  | The Bully timestamp is a divisor of Earth's sidereal year.
  | The Sun orbits approximately one solar radius per Bully timestamp.
  }}
}}

==== Earth's tropical year ====
Earth's tropical year, which measures the complete cycle of seasons between successive vernal equinoxes, spans '''31,556,925.2 seconds'''. Due to axial precession, the tropical year is roughly 2/5 of a Bully timestamp shorter than the sidereal year.

==== Earth's Great Year ====
With Earth's sidereal year (&lt;math&gt;P&lt;/math&gt;) spanning 10,330 timestamps and the tropical year (&lt;math&gt;a_{t}&lt;/math&gt;) spanning 10,329.6 timestamps, a full precessional cycle requires a number of years, &lt;math&gt;N&lt;/math&gt;, where the cumulative annual difference equals exactly one year:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
N &amp;= \frac{1}{10,330 - 10,329.6} \\
&amp;= \frac{1}{0.4} \\
&amp;= \frac{5}{2}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Expressing this duration in terms of sidereal years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,329.6 \, P \\
&amp;= 25,824 \, P 
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

Alternatively, expressing the cycle in terms of tropical years yields:

&amp;hairsp;
&lt;math&gt; 
\begin{aligned}
1 \, \text{Great Year} &amp;= \frac{5}{2} \times  10,330 \, a_{t} \\
&amp;= 25,825 \, a_{t}
\end{aligned}
&lt;/math&gt;
&amp;hairsp;

The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps.

* [[Bully Mnemonic |Learn More About The Bully Mnemonic]]

==== Earth's Seasons and Milky Way Visibility ====
In 1998 in the Northern Hemisphere, winter lasted 89 days, spring lasted 92 days and 18 hours, summer lasted 93 days and 15 hours, and autumn lasted 89 days and 21 hours. Summer was nearly five days longer than winter that year. As shown in '''Figure 5c''', this duration discrepancy will continue to increase for the next 1,500 years until summer is a full 94 days long and winter is less than 89 days.

The Earth's orbital speed varies throughout the year, moving slowly during [[W:aphelion|aphelion]] and quickly during [[W:perihelion|perihelion]]. Consequently, whichever season is aligned with aphelion ends up being the longest because the Earth is moving slowly and takes longer to get through that season.

As shown in Figure 5c, '''winter''' was the longest season in the Northern Hemisphere (aligned with aphelion) back before 5,000 BCE. It took approximately 5,250 years to cycle to '''spring''' being the longest season, and another 5,250 years to '''summer'''. While it is just beyond the range of the graph, it is clear that all four seasons will complete a full cycle once in a little over '''21,000 years'''.

[[File:Earth_Seasons_and_Milky_Way_Visibility_Shifts_Over_Time.svg|thumb|center|800px|alt=Graph showing how the lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.|'''Figure 5c''': The lengths of Earth's seasons, and the season with best Milky Way visibility, shifts over time.]]

Currently, the Milky Way is easy to view during Northern Hemisphere summer months (which are winter months in the Southern Hemisphere). Going back in time prior to 1998, there was an era when the Milky Way would have been more visible during spring months in the north. Going even further back, the Milky Way would have been best viewed in northern winter (or southern summer). These seasonal shifts in Milky Way visibility are correlated with large dots in Figure 5c. 

The transition from '''spring to summer''' is correlated with a large '''green dot''' and a green banner indicating that astronomer Jean Meeus identified May 1998 CE as the precise moment when the Galactic Equator crossed the solstice points. A large '''red dot''' appears in 8329 CE to indicate the approximate crossing from '''summer to autumn''', and a large '''blue dot''', back in 4495 BCE, indicates the approximate crossing from '''winter to spring'''. The red and blue dots represent epochs when the coordinate latitude of the Sun, as viewed from Earth, is nearly zero in the ecliptic, celestial, and galactic coordinates at the same time. These large dots representing Galactic Equator crossings occur about once every 6,500 years. While it is beyond the range of the graph, it is clear that the Milky Way passes through all four seasons during a time period of roughly '''26,000 years''' (1 Great Year ≈ 25,824 sidereal years ≈ 25,825 tropical years).

{{Quote box | align = center | width = 100% | title = Bully Time Anchor | quote = The time anchor of the entire Bully system was selected to be '''12:00:00 TAI on June 21, 1998''', which is near the 1998 June solstice. This date was selected as the anchor because it occurred in the month following the May 1998 CE date that Jean Meeus identified as a Galactic crossing. All of the Galactic crossings shown in Figure 5c, at 4495 BCE, 1998 CE, and 8329 CE, are roughly correlated with rounded Bully timestamps.
&lt;ol&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 E900 0000}}''' aligns with 4500 BCE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 ED00 0000}}''' aligns with 1998 CE.&lt;/li&gt;
  &lt;li&gt;Bully timestamp '''{{nowrap|8209 F100 0000}}''' aligns with 8494 CE.&lt;/li&gt;
&lt;/ol&gt;
}}

==== The Metonic cycle ====
The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp complete approximately three cycles per one Metonic cycle as illustrated in the following list:
&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
July 23 New Moons
* July 23, 1998 — 8209 ED0'''0 038B'''
* July 23, 2017 — 8209 ED0'''3 0238'''
* July 23, 2036 — 8209 ED0'''6 00EA'''
&lt;/div&gt;


[[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]]

== Bully Timestamp Realization ==
Each Bully timestamp is '''realized''' exactly 3055 seconds TAI after the previous one.  However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an '''estimate''' of how time might have transpired in the past, rather than an actual realization of Bully time. Similarly, any assignment of future timestamps should be viewed as an estimate of what may occur, rather than a realization. Bully timestamps should only be considered &quot;realized&quot; when time is measured with an accuracy of 10&lt;sup&gt;-10&lt;/sup&gt;. There have been over 700,000 realized Bully timestamps during the era of modern atomic time keeping (1958 AD ... present).

[[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]]

== Bully Timestamp Estimation ==
[[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 6a: History of the Universe with a few example Bully timestamps shown in red.]]
For the purpose of time estimation, the Bully system's time range is divided into three distinct sets:

==== First Set ====
* ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Used to estimate time during the universe's formative period ('''Figure 6a'''), spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events from selected timestamps during this formative era:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* First timestamp: ''{{mono|0000 0000 0000}}''
** [[w:Cosmic_inflation|Cosmic Inflation]]
** [[w:Baryogenesis|Baryogenesis]]
** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]]
* Approximately: ''{{mono|0000 EA00 0000}}''
** [[w:Decoupling_(cosmology)|Decoupling]]
** [[w:Recombination_(cosmology)|Recombination]]
* Approximately: ''{{mono|0100 0000 0000}}''
** [[w:Star_formation|First Star Formation]]
* Approximately: ''{{mono|0297 0000 0000}}''
** [[w:MoM-z14|Oldest Observed Galaxy]]
&lt;/div&gt;

==== Second Set ====
* ''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'': Used to estimate cosmic look-back time ('''Figure 6b'''), spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones from the presolar through geological eras include:
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|3B00 0000 0000}}''
** [[w:Murchison_meteorite|Oldest Presolar Grains]]
* Approximately: ''{{mono|5720 9000 0000}}''
** [[w:Hadean|Hadean Eon Begins]]
* Approximately: ''{{mono|5C2A 0000 0000}}''
** [[w:Archean|Archean Eon Begins]]
* Approximately: ''{{mono|6A8C 0000 0000}}''
** [[w:Proterozoic|Proterozoic Eon Begins]]
* Approximately: ''{{mono|7D56 0000 0000}}''
** [[w:Phanerozoic|Phanerozoic Eon Begins]]
&lt;/div&gt;

[[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 6b: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]]
 (Ma) represents one million (10&lt;sup&gt;6&lt;/sup&gt;) years.]]

==== Third Set ====
* ''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'': Used to estimate (and realize) future events. This set begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years.
&lt;div style=&quot;background-color: #f0f4f7;{{Text color default}}; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
* Approximately: ''{{mono|B000 0000 0000}}''
** [[w:Sun#Life_phases|Death of Sun (main-sequence)]]
&lt;/div&gt;

=== Time Estimation Using Cosmic Redshift ===
In [[w:physics|physics]], a '''redshift''' is an increase in [[w:wavelength|wavelength]] (or a decrease in [[w:frequency|frequency]]) of [[w:electromagnetic radiation|electromagnetic radiation]]. Cosmological redshifts are driven directly by the [[w:expansion of the universe|expansion of the universe]]. The redshift value is denoted by {{math|''z''}}, where the ratio of observed to emitted wavelength is {{math|1 + ''z''}}.

If the original wavelength of a radiation source is known, its cosmological redshift can reveal the light travel time. However, mapping redshift precisely to elapsed time requires an exact cosmological model. Ongoing measurement tension surrounding the [[w:Hubble constant|Hubble constant]] introduces uncertainty into calculations of the exact [[w:Age of the universe|age of the universe]] and distant stars.

This cosmological uncertainty directly affects the accuracy of assigning Bully timestamps. The table in Figure 6c contrasts two estimation tracks based on competing cosmological datasets. One column applies the local distance ladder framework from the '''SH0ES Team''' (corresponding to a younger universe estimate of 12.7 Gyr). The other utilizes cosmic microwave background data from the '''Planck Collaboration''' (yielding an older universe estimate of approximately 13.8 Gyr). Larger z values correspond with the more distant past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6c: Bully Timestamps for Selected Redshift Values Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0; font-size: medium; font-weight: bold;{{Text color default}};&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Redshift z &lt;br /&gt; (z = ∞ to 2)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};”&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = ∞ || {{nowrap|0000 0000 0000}} || {{nowrap|0000 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 18.0 || {{nowrap|01CC 0000 0000}} || {{nowrap|01F4 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 15.0 || {{nowrap|0253 0000 0000}} || {{nowrap|0287 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 12.0 || {{nowrap|032D 0000 0000}} || {{nowrap|0374 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 9.0 || {{nowrap|04B5 0000 0000}} || {{nowrap|051E 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 6.0 || {{nowrap|0809 0000 0000}} || {{nowrap|08BB 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | z = 3.0 || {{nowrap|1285 0000 0000}} || {{nowrap|1420 0000 0000}}
|- style=&quot;font-size:small:small;background-color:#ffffff;;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;;{{Text color default}};&quot; | z = 2.0 || {{nowrap|1C4D 0000 0000}} || {{nowrap|1EC2 0000 0000}}
|}

The forward-progressing timestamps ''{{mono|0000 0000 0000}}'' through ''{{mono|1FFF FFFF FFFF}}'' are illustrated in Figure 6d (bottom of figure). By convention, these timestamps are assumed to begin at the Big Bang and progress forward for approximately three billion years.

[[File:Redshift-by-universe-age-H0-comparison.png|frame|center|alt=Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.|Figure 6d: Age of the Universe plot showing Bully timestamps mapped to cosmic redshift.]]

Timestamps ''{{mono|2000 0000 0000}}'' through ''{{mono|8200 0000 0000}}'' (top of Figure 6d) measure &quot;lookback&quot; time anchored at timestamp ''8209 ED00 0000''. Because the total age of the universe is unfixed, the precise mathematical relationship between universal age and lookback time remains indefinite. Two different possible universe ages are shown with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

The data illustrated in Figure 6e is the same as is shown in Figure 6d, but Figure 6e plots against lookback time on the x-axis, so in this plot the universe age is unfixed with the Planck Collaboration shown in red and the SH0ES Team shown in blue.

[[File:Redshift-by-lookback-time-H0-comparison.png|frame|center|alt=A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.|Figure 6e: A cosmic lookback plot showing Bully timestamps mapped to cosmic redshift.]]

The table in Figure 6f is similar to the table in Figure 6c, in that it contrasts two estimation tracks based on competing cosmological datasets. However, whereas the data in Figure 6c was for large z values, Figure 6f shows small z values. Smaller z values correspond with the recent past. 

{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;&quot;
|+ Figure 6: Redshift Values for Selected Bully Timestamps Given Different Universe Age Estimates

|- style=&quot;background-color: #eaecf0;{{Text color default}}; font-size: medium; font-weight: bold;&quot;
! style=&quot;padding: 10px; font-size: large;&quot; | Bully Timestamp &lt;br /&gt; (z = 1 to 0)
|| SHOES Team &lt;br /&gt; (12.7 Gyr) || Planck Collaboration &lt;br /&gt; (13.8 Gyr)
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|4000 0000 0000}} || z = 0.925134 || z = 0.796535
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|6000 0000 0000}} || z = 0.342787 || z = 0.308619
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8000 0000 0000}} || z = 0.016418 || z = 0.015093
|- style=&quot;font-size:small:small;background-color:#ffffff;{{Text color default}};&quot;
| style=&quot;font-weight: bold; background-color: #eaecf0;{{Text color default}};&quot; | {{nowrap|8209 ED00 0000}} || z ≈ 0.000000 || z ≈ 0.000000
|}

===  Time Estimation Relativistic and Cosmological Considerations ===
What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference?

The &quot;age of the universe&quot; cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the &quot;CMB rest frame.&quot;

Importantly, Bully timestamps are divided into three distinct sets, with only the first set (''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'') utilizing the CMB rest frame. Timestamps in the third set (''{{mono|8209 ED00 0000}}'' — ''{{mono|FFFF FFFF FFFF}}'') are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. All &quot;realized&quot; Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference.

Furthermore, the &quot;estimated&quot; Bully timestamps in the second set (''{{mono|2000 0000 0000}}'' — ''{{mono|8209 ED00 0000}}'') are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decay at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited.

[[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]]

== Contextualized vs. Decontextualized Time ==

Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is &quot;contextual&quot; because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic &quot;leaps&quot; (seconds and years). In '''Figure 10''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1.

In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to &quot;true time&quot; at any specific geographical location. Represented by the black lines in '''Figure 10''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures.

[[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 10: Modern Time Keeping]]

The various decontextualized standards currently in use are effectively &quot;frozen&quot; in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds.

The Bully timestamp system, shown on the far-right axis of '''Figure 10''', follows the same uniform, decontextualized logic as TAI and TT but avoids this &quot;legacy offset&quot; confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret.

[[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]]

== Why do we need Bully timestamps? ==

All the timestamps in '''Figure 11''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. 

{| class=&quot;wikitable&quot; style=&quot;margin-right: 0; margin-left: 1em; text-align: center;&quot;
|+ Figure 11: UTC Time Zones vs. Bully Timestamps.
|-
! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps]
|-
| rowspan = 3 |
[[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0|
June 21, 1998 at  8:59:29 pm (JST)&lt;/br&gt;
June 21, 1998 at  7:59:29 pm (CST)&lt;/br&gt;
June 21, 1998 at  2:59:29 pm (EEST)&lt;/br&gt;
June 21, 1998 at 12:59:29 pm (IST)&lt;/br&gt;
June 21, 1998 at 11:59:29 am (GMT)&lt;/br&gt;
June 21, 1998 at  8:59:29 am (BRT)&lt;/br&gt;
June 21, 1998 at  4:59:29 am (PDT)&lt;/br&gt;
June 21, 1998 at  1:59:29 am (HST)&lt;/br&gt;
]]
||
[[File:WorldMap-Blank-Noborders.svg|thumb|&lt;br/&gt;
06/21/1998 12:00:32.184 (TT)&lt;br/&gt;
06/21/1998 12:00:00 (TAI)&lt;br/&gt;
06/21/1998 11:59:42 (GPS)
]]
|-
! Bully Timestamp
|-
||
[[File:WorldMap-Blank-Noborders.svg|thumb|8209 ED00 0000 (+ 0.000 sec)]]
|}

==== Legacy Decontextualized Timestamps ====

The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 11''' attempt to solve the UTC geographic fragmentation problem, yet they remain &quot;cluttered&quot; by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several &quot;leap&quot; seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time.

For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains &quot;tethered&quot; to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome &quot;hybrids&quot; that attempt to measure linear time using units designed for Earth’s rotation.

==== Decontextualized Bully Timestamps ====

The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 11''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 ED00 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time.

Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format:

[http://www.leapsecond.com/m/gps.htm LeapSecond.com]

[https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com]

[http://www.csgnetwork.com/multitimedisp.html csgnetwork.com]

== The Foundations of Bully Metric ==

The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to &lt;math&gt;10,330 \times 3,055&lt;/math&gt; SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system.

The name &quot;Bully&quot; is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, &quot;bully&quot; means '''&quot;beautiful&quot; or &quot;excellent,&quot;''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of &quot;bullies&quot; like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system.

* [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]]
* [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]]</text>
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|&lt;div style=&quot;font-size:200%;border:none;margin: 0;padding:.1em;color:#000&quot;&gt;'''International Communication Law Learning Project'''&lt;/div&gt;&lt;div style=&quot;top:+0.2em;font-size:150%&quot;&gt;part of the [[School:Law|'''School of Law''']]&lt;/div&gt;
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{{SearchWithPrefix}}'''[[Help:Resources by subject|Subject classification]]''': this is a [[Portal:Law|law learning projects]] resource.

'''Class''': International Communication Law

'''Professor''': Juan Carlos Riofrio
[[File:International Communication Law.png|thumb|300x300px]]Welcome to the '''International Communication Law''' Learning Project, part of the [[School:Law|School of Law]]. This learning project aspires to establish the highest standards in the field of international communication law by fostering a comparative analysis of legal systems across different countries. By examining and contrasting the regulations, principles, and practices governing communication in diverse jurisdictions, the project aims to identify best practices and promote a deeper understanding of global legal standards.

The ultimate goal of this project is to equip legal professionals with the knowledge necessary to navigate the complexities of cross-border communication, ensuring compliance with diverse legal systems while upholding fundamental principles of fairness and accuracy.

== 2025 ==

[[Armenian Legal System]]

[https://w.wiki/F6Yy France's Legal System]

[[Italian Legal System|Italian Legal System]]

[[Canadian Communications Law|Canadian Legal System]]

[[Saudi Arabian Legal System|Saudi Arabian Legal System]]

[[United Kingdom Legal System]]

[[Puerto Rico's Legal System]]

[[German Legal System]]

[[Russian Legal System]]

[[Jamaican Legal System]]

[[Brazilian Legal System]]

[[Communications Law in Austria]]

[[Republic of Ireland Legal System]]

[[Australia's Legal System]]

== 2026 Spring ==
[[Communications Law in Malta]]

[https://w.wiki/Hc5V The Kingdom of Morocco]  

[[Communications Law in China]]

[[The Netherlands' Legal System|Communications Law in the Netherlands]]

[[Communications Law in India]]

[[Communications Law in Portugal]]

[[Media Law in the Czech Republic]]

[[Communications Law in Spain]]

Communication Law in [[Communication Law in Spain|Spain]]

[[Communications Law in Mexico]]

== 2026 Fall ==
[[Communication Law in Ecuador]]

[[Communication Law in New Zealand]]

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      <text bytes="32345" sha1="rwegozhlezgf1qzgza0119i6u92tsur" xml:space="preserve">{{title|Self-determination theory and social media use:&lt;br&gt;How do basic psychological needs explain patterns of social media engagement?}}__TOC__

==Overview==
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[[File:Social media icons (rubin-Social-media-icons-2x).tiff|thumb|150px|'''Figure 1.''' Social media networks. ''(Author note: Alternatively use image of person using social medi''|right]]

'''Consider this scenario'''

You’re taking a quick study break and plan to make a cup of tea. While the kettle boils, you hear a ding and it is an Instagram notification showing that your friend has sent you a message. You then reply to that friend feeling briefly connected (see Figure 1). You spot a reel on psychology and self-determination theory and choose to watch, telling yourself it’s still related to your studies. But then the algorithm pulls you in. Psychology reels lead to study hacks, then lifestyle tips, then memes, then sheep jumping in fields. You come across a scrolling check-in video and quickly swipe down to keep scrolling. You stop noticing what you're choosing. Thirty minutes pass. Your tea is cold. You started in control, with purpose and feeling connected. Now it’s unclear what you were looking for and you feel less motivated to study. You didn’t mean to spend so long. Where did the time go?

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This chapter explores [[w:Self-determination_theory|self-determination theory]] (SDT) and how basic psychological needs (BPNs) according to [[Motivation and emotion/Book/2020/Basic psychological need theory|basic psychological need theory]] (BPNT) can explain patterns of social media engagement. Social media is becoming increasingly intertwined with (deeply &lt;u&gt;embedded in&lt;/u&gt;) everyday life, influencing how people communicate, relate &lt;u&gt;(to one another),&lt;/u&gt; and see the world and themselves/present themselves &lt;u&gt;--&gt; and at a fundamental level how people think, feel, and act/behave&lt;/u&gt; (&lt;u&gt;ref&lt;/u&gt;). 

* Pervasiveness of social media. Social media is deeply embedded in everyday life, influencing how people connect, communicate, and present themselves.
* The problem. While social media offers benefits like connection and self-expression, it is also linked to negative outcomes such as anxiety, low self-esteem, and poor well-being, particularly with excessive or unbalanced use.
* The need for psychological science. Understanding why people engage with social media and how it affects their mental health requires a psychological science perspective with relevant theories and evidence explaining the link between social media, motivation and well-being.
* Introduction to Self-Determination Theory (SDT). SDT is a well-established motivation theory that helps explain human behaviour by focusing on the satisfaction of three core psychological needs.
* Basic Psychological Need Theory (BPNT). A mini-theory of SDT, BPNT identifies three universal psychological needs: Autonomy (the need to feel in control), competence (the need to feel capable and effective), and relatedness (the need to feel socially connected and a sense of belonging) (Vansteenkiste et al., 2020). These BPNs are drivers of human behaviour and motivation.
* How social media interacts with these needs. Social media can support these needs. It can also frustrate them (e.g. comparison or social pressure)
* The role of design. Social media platforms are intentionally designed to engage users by targeting these psychological needs
* Why this matters. Applying SDT offers a deeper understanding of social media engagement and highlights ways to promote healthier, more intentional use that supports well-being.
* [[psychological needs]]
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'''Focus questions'''

* What is the relationship between social media, motivation and well-being?
* What psychological processes underpin social media design and patterns of engagement?
* What is self-determination theory?
* What are basic psychological needs according to self-determination theory?
* How do basic psychological needs explain patterns of social media engagement?
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== Social media, motivation and well-being ==
'''Focus question''': What is the relationship between social media, motivation and well-being?

Global prevalence of social media and integration into daily lives. The evolving landscape of digital technology including social media, also called social network sites (SNS), means virtual realms are increasingly integrated into everyday life and create new pathways for engagement and interaction. Excessive social media use poses significant concerns and consequences at individual and societal levels.

Distinguish between social media use and [[w:Problematic_social_media_use|problematic social media use]] which is associated with internet addiction disorder, a form of behavioural addiction (ref).

* Social media use is directly linked to screentime use as well as smartphone use. Statistics show social media is primarily accessed via smartphones (references). To find out more check out this chapter on [[Motivation and emotion/Book/2025/Mobile phone use motivation|mobile phone use motivation]].
* ''Author note: Expand on or link to issues of social media engagement and patterns of engagement. Focus on how social media impacts wellbeing and the psychological impacts of social media use.''
* There is a growing body of research examining the relationship between social media and well-being with motivation a key element linking drivers of social media engagement. Need for research using theoretical frameworks.
* Social relationships are at the heart of social media. These platforms were designed to help people communicate, stay in touch with friends and family, and share information. While their original purpose may have been to support social connection, social media has since evolved into a powerful tool with an attention-based based business model (Montag et al., 2019). ''Aurthor note check ref if 2019 0r 2021''
* Studies have shown social media can have both a positive and negative impact on well-being (Parent, 2023; West et al., 2024). Understanding what motivates people to use social media is critical to understanding how social media impacts well-being and what can be done to promote healthy social media use.
* [[Motivation]] is the energy for action (Ed Deci...)... quality of behavior not quantity of behavior

* The average person spends 145 minutes per day on social media (ref).
* Short-form videos are the most engaging content form on social media often being under 1 minute in length, capture attention due to bite-sized format (ref)
* Majority social media users access social media via mobile phones, increasing social medias accessibility and prevalence in everyday life.
* Demographic youth and young adult's social media usage skewed toward younger demographics with 
* &quot;Correlation between social media and depression for adolescents&quot; Time spent e.g. &quot;Adolescents who spend more than 3 hours a day have an increased risk of mental health struggles.&quot;
* Intrinsic vs extrinsic motivation --&gt; link to BPNs and intrinsic motivation
== The psychology behind social media design and patterns of engagement ==
[[File:Social media integration and central role in everyday life - media networks and connection.png|thumb|350x350px|'''Figure 2.''' Intergration and central role of social media in everyday life. Social media is increasingly becoming interwoven with everyday life, social connection and belongin]]'''Focus question:''' What psychological processes underpin social media design and patterns of engagement?

* Understanding the psychology behind social media design is important as it informs patterns of social media engagement and the psychological processes social media is designed to act on.
* ''Author note: Research more into psychological processes and possibly principles behind social media design and that are targeted when people use social media. While potentially not necessary, this section provides the opportunity to expand on the problem, how social media is closely, possibly intrinsically, linked to psychology and psychological science, briefly look at how other theories or aspects of motivation relate to social media, and lead into why self-determination theory and BPNT is a suitable (or the best/ideal) lens through which explain patterns of social media engagement.''

=== Social media design ===
* Social media is intentionally designed to capture one's/people's attention and hold it for as long as possible and includes addictive features (Montag et al., 2019).
* Also consider physiological and neurological processes that social media is designed to engage e.g. dopamine cycle
* Reward-seeking, social validation etc.
* Discuss theoretical and psychological underpinnings behind social media design and how it connects with psychological science, self-determination theory, BPNT, motivation and therefore why psychology can be used to understand engagement patterns. --&gt; ''potentially move above self-determination theory section''
* (see Figure 2)

=== Patterns of social media engagement ===

* Social media engagement is multi-dimensional and concerns affective, cognitive, and behaviour dimensions (Trunfio and Rossi, 2021).
* The Consumer Online Brand Related Activities Model (COBRA) is a conceptual tool used to classify and interpret social media engagement with brands and differentiates between three levels: consumption, contribution and creation (Schivinski et al., 2016). These levels of social media engagement encompass various actions including viewing, liking, sharing, commenting, and content creation (Trufino &amp; Rossi, 2021).
* Expand on problematic social media use compared to healthy patterns of social media engagement.
* ''Author note: Potentially split this section between sections: Social media, motivation and well-being AND basic psychological needs and social media engagement.''

== Self-determination theory ==
'''Focus question:''' What is self-determination theory?[[File:Self-Determination-Theory-Visual 1.png|thumb|'''Figure 3.''' Self-determination theory.|400x400px]][[Self-determination theory]] (SDT) is a key theory of [[w:Motivation|motivation]], playing a central role in [[w:Psychology|psychology]] and empirical studies on motivation (Ryan et al., 2021). SDT highlights human capacities and innate resources for behavioural self-regulation and personality development (Ryan et al., 2021).
* A key component of self-determination theory is the sub-theory basic psychological need theory (see Figure 3).
* SDT was first developed by Ryan and Deci and has since been widely used in research and evidence-based application across diverse fields such as health care, education, work, sport, technology use, psychotherapy, parenting, developmental and organisation psychology (Ryan et al., 2021).
* SDT is a well-established and robust theory, recognised for its universality/as a universal theory across cultures and domains (Ryan et al., 2021). Multi-dimensional wellbeing framework.
* SDT divides motivation into two categories: [[w:Intrinsic_motivation|intrinsic motivation]] and [[w:Extrinsic_motivation|extrinsic motivation]] (Ryan et al., 2021).
* Autonomous (intrinsic and extrinsic motivation) and controlled motivation.

* Possibly include figure of &quot;Self-determination theory's taxonomy of motivation&quot; (see Figure 4).
* Motivation is the and is related to activation and intention (Ryan &amp; Deci, 2000).
* SDT is comprised of six mini-theories: cognitive evaluation theory (CET), organismic integration theory (OIT), causality orientations theory (COT), basic psychological needs theory (BPNT), goal contents theory (GCT), and relationships motivation theory (RMT) (Ryan &amp; Deci, 2018). ''Aurthor note: Check ref 2017 or 2018.''
[[File:Amp-55-1-68-fig1a.gif|center|thumb|758x758px|'''Figure 4.''' Self-determination theory's taxonomy of motivation. ]]

== Basic psychological need theory ==
'''Focus question:''' What are basic psychological needs according to self-determination theory?

Basic psychological need theory (BPNT) is a mini theory/sub-theory within the SDT framework (Vansteenkiste et al., 2020). BPNT posits humans have three basic psychological needs: autonomy, competence, and relatedness (Ryan &amp; Deci, 2000) (see Figure 5). Psychological needs are critical resources underlying the inherent desires to engage with the environmental to advance personal growth, social development, self-organisation, and psychological well-being (Reeve, 2018; Vansteenkiste et al., 2020). According to BPNT BPNs are universal and must be satisfied for &lt;u&gt;(are crucial to)&lt;/u&gt; optimal well-being and performance, and if unsatisfied leads to negative consequences such as ill-being (Ryan et al., 2021). BPNT emphasises the importance of creating optimal circumstances to support peoples' basic psychological needs, with social and sociocultural contexts being a key/the main avenue for nurturing or thwarting needs (Vansteenkiste et al., 2020). BPNs drive human behaviour being intrinsically linked to motivation. 
According to BPNT, a need must meet nine conceptual and empirical criteria. The basic criteria dictate a need must be psychological, essential, inherent, distinct, and universal. The associated criteria state a need must also be pervasive, content-specific, directional, and explanatory (Vansteenkiste et al., 2020).[[File:Basic needs.png|thumb|'''Figure 5.''' Theory of basic psychological needs. &quot;'''Figure #'''. The three basic psychological needs proposed by self-determination theory are considered essential ingredients for intrinsic motivation and psychological well-being.&quot;]]
* Ryan and Deci (2000) define basic needs as energising states which when satisfied contribute toward health and well-being, however if unsatisfied lead to ill-being.
* BPNT initially
* &quot;Drivers of human action essential for psychological growth and activity&quot;
* Basic needs drive behaviour in order to satisfy/fulfill said needs.
* &quot;Basic psychological needs important motivational tools...&quot;
* BPNT relevant in social and non-social contexts.
* Universal psychological nutrients (Ryan &amp; Deci, 2018).
* &quot;Intrinsic motivation is fostered by conditions which support autonomy, competence, and relatedness (Lecture 4)&quot;
* BPNs cross-culturally (Grubbs et al., 2025)
=== Autonomy ===
Autonomy refers to the psychological need to experience willingness, volition, and self-organisation (Ryan et al., 2021; Vansteenkiste et al., 2020).

* &quot;...refers to the capacity to make informed and independent decisions.&quot;

* Autonomy is satisfied when actions, thoughts and feelings are self-endorsed and authentic creating the experience of integrity (Vansteenkiste et al., 2020).
* Autonomy is frustrated when conflict or pressure is felt that opposes one volition (Vansteenkiste et al., 2020).
* Autonomy support: their perspective, self-initiation, choice, explore, and rationale. 

=== Competence ===

* Competence is the psychological need to experience effectiveness and mastery (Vansteenkiste et al., 2020).
* &quot;...ability to do task effectively and efficiently.. improves task performance --&gt; task mastery. 
* Competence is satisfied when one is able to utilise and develop skills and expertise (Vansteenkiste et al., 2020). &quot;...engaging in activities on can make use of their skills and expertise.&quot;
* Competence is frustrated by experiences of failure, ineffectiveness, and helplessness (Vansteenkiste et al., 2020).
=== Relatedness ===

* Relatedness refers to is the psychological need to experience bonding, care, and warmth from others (Vansteenkiste et al., 2020).
* &quot;..need for connection, satisfied through bonding and care.&quot;
* Relatedness is closely tied to relationships and social connection and is satisfied by feeling connected to others (Vansteenkiste et al., 2020).
*Relatedness is frustrated by social exclusion, social alienation, and loneliness (Vansteenkiste et al., 2020).

== Basic psychological need satisfaction and frustration ==
* Expression of basic psychological needs varies across cultures due to the internalisation and integration of cultural values and behaviours facilitating need satisfaction (Ryan &amp; Deci, 2000).
* Need satisfaction is foundational for wellness and healthy functioning (Ryan et al., 2021).
* Need frustration is detrimental to well-being and diminishes integrity (Ryan et al., 2021).

=== Need satisfaction and need support ===
* SDT and BPNT are primarily concerned with fostering human growth and wellness through need satisfaction and need support (Vansteenkiste &amp; Ryan, 2013).
* The satisfaction of basic needs is facilitated by need supportive environments and promotes wellness, the development of inner resources, and resilience (Vansteenkiste &amp; Ryan, 2013).
* Satisfaction's role in motivation is to energise and sustain intrinsic motivation and internalisation (Ryan et al., 2021).

=== Need frustration and need thwarting ===
* Need frustration and need thwarting are associated with negative outcomes such as ill-being, self-control breakdowns, constricted functioning, externalising problems and increased defensive and immoral functioning (Vansteenkiste &amp; Ryan, 2013).
* &quot;...emotional response to opposition.&quot; 
* Need frustration and need thwarting are worse than the absence of need satisfaction and need support (Vansteenkiste &amp; Ryan, 2013).
* Needs frustration diminishes the energisation and sustenance of intrinsic motivation and internalisation, and elicits defiance (Ryan et al., 2021).
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- Autonomy

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== Basic psychological needs and social media engagement ==
'''Focus question:''' How do basic psychological needs explain patterns of social media engagement?

* BPNT can help explain patterns of social media engagement... (ref).
* Social media creates unique social contexts which
* Different factors of social media engagement can contribute to basic psychological need satisfaction and frustration including intra-personal, inter-personal, situational and environmental factors (West et al., 2024)
* The first scoping review examining the relationship between social media use and well-being in adolescents through a SDT lens found social media use can be both beneficial and detrimental to satisfying basic psychological needs of autonomy, competence and relatedness (West et al., 2024). Figure # represents how engagement with social media can satisfy BPNs supporting well-being, performance, and growth, however unfulfilled needs are not conducive for individuals to thrive.

*Table 1 shows examples of
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[[File:Social media use impact girls mental health plants thrive or wilt raw.png|left|thumb|350x350px|'''Figure 6.''' Impact of social media use on mental health and basic psychological needs. Plants represent thriving and fulfillment of needs vs frustration of needs ''([[ccorg:licenses/by-sa/4.0|Mercy Canning, CC BY-SA 4.0]], via Wikimedia Commons).'']]

=== Autonomy and social media engagement ===

* Social media supports autonomy by creating a ''(enabling one to experience a)'' sense of control, self-governance and personal agency (West et al., 2024).
* Social media has the potential to influence autonomy through identity development (West et al., 2024).
* Social media can challenge and threaten autonomy as choices and actions may be constrained or limited and individuals may be susceptible to other people's actions such as controlling behaviours (West et al., 2024).
* Expand on need satisfaction and support for autonomy
*Expand on need frustration and thwarting for autonomy
*Passive social media use is associated with lower autonomy and ill-being (ref)
*(see Figure 6)
=== Competence and social media engagement ===

* Sense of self can be positively or negatively impacted by social media (West et al., 2024). Areas related to sense of self that may be influenced by social media include self-esteem, confidence, self-concept, social competence... 
* Other areas impacted by social media include learning, digital competence (West et al., 2024). 
* The impact of social media on competence is consistent with other basic psychological needs in that social media may support or thwart competence (West et al., 2024)
* A study by West, Rice and Vell-Brodick (2023) found social media contributes to adolescent's experience of unique challenges regarding competence and identified two broad themes: cultivating competence and constraining competence.
* Expand on need satisfaction and support for competence
*Expand on need frustration and thwarting for competence
[[File:Social media positive &amp; negative effects on mental health &amp; well-being.png|thumb|350x350px|'''Figure 7.''' Description.]]

=== Relatedness and social media engagement ===

* Social media plays a key role in satisfying and frustrating the basic psychological need of relatedness (West et al., 2024). 
* Social media may be used to experience a sense of belonging and acceptance (West et al., 2024). 
* Social media enables parasocial relationships. Social media can support well-being when used for self-disclosure.

*Expand on need satisfaction and support for relatedness
*Expand on need frustration and thwarting for relatedness
*(see Figure 7)
*(see Table 1) organise and summarise info 
'''Table 1.''' 

Basic Psychological Needs Role in Motivation and Influence on Social Media Patterns of Engagement 
{| class=&quot;wikitable&quot;
!Basic psychological need
!Role in motivation
!Applied to social media engagement
|-
|[[w:Autonomy|'''Autonomy''']]
|
|
|-
|'''[[w:Competence|Competence]]'''
|
|
|-
|'''Relatedness'''
|Social connection

*

*
|
|}
''Author note: Either incorporate need satisfaction and frustration under related basic need level 1 sub-headings as two separate paragraphs (with or without level 2 sub-heading) OR create new major heading addressing need satisfaction and frustration. However, basic psychological needs explaining social media use naturally addresses satisfaction and frustration while explain how each basic need is related to social media. Consider structuring paragraphs under level 1 sub-heading by topic rather than needs satisfaction and frustration e.g. different types of competence social media influences such as sense of self and digital competence.''

''Author note:'' Potentially add case studies or examples of patterns of social media use explained by basic psychological needs for key age groups across the lifespan. Alternatively, case studies or examples in different environments/contexts.

* Case study on youth/adolescents
* Case study on university students
* Case study on adults or elderly adults


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== Promoting health and well-being through social media engagement ==
''Author note: Potentially add additional major heading and section focused on action orientated tips for healthy social media use that satisfies and supports basic needs and wellness.''

* Alt title: How to engage with social media to promote health and well-being through satisfaction of basic psychological needs

* Limiting social media use
* Use social media as an avenue for connection with friends and family while also prioritising connection with others in real-life face-to-face ways.
* Setting clear goals and motives for social media use
* Using social media as a creative outlet
* If you get struck
* '''Potential solutions?''' &quot;Limiting social media time, increasing media literacy and redesigning platforms might be keys to healthier digital interactions to counter this troubling trend. (Forbes).&quot;

==Conclusion==
* Social media use is a prevalent part of daily life for many people. Linked to increase in technology.
* SDT, specifically BPNT, helps explain social media patterns of engagement.

* Social media is deeply embedded in everyday life, offering opportunities to connect, express, and engage, but it also brings significant risks to mental health and well-being. 
* Psychological science, particularly SDT and BPNT, helps explain the complex relationship between social media, motivation, and well-being. According to BPNT, all humans have three basic psychological needs which are essential for psychological wellness and optimal performance: autonomy, competence, and relatedness. 
* Social media can support these needs by allowing users to make choices, build skills, and feel connected. However, it can also frustrate or thwart these needs, such as when users experience pressure, comparison, exclusion, or lack of control.
* The psychological processes behind social media design act on these basic needs and influence patterns of engagement that may feel rewarding but can become problematic over time. 
* Understanding underlying mechanisms of social media can help individuals and society make more informed, intentional decisions about how to engage with social media.
* Rather than framing social media as entirely good or bad, SDT provides a lens through which to understand its effects. 
* Supporting autonomy, competence, and relatedness is key to promoting healthy, self-determined social media use and enhancing overall well-being in a digital world.
* (see Figure 8)


''Author note: Address these points in conclusion''
* ''Answer to sub-title question based on psychological theory and research?''
* ''Answers to focus questions?''
* ''Practical, take-home messages?''
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* Message{{RoundBoxBottom}}
==See also==
* [[Motivation and emotion/Book/2025/Autonomy and intrinsic motivation in self-determination theory|Autonomy and intrinsic motivation in self-determination theory]] (Book chapter, 2025) 
* [[Motivation and emotion/Book/2020/Basic psychological need theory|Basic psychological need theory]] (Book chapter, 2020) ''Author note: TBC keep''
* [[Motivation and emotion/Book/2021/Frustration of basic psychological needs|Frustration of basic psychological needs]] (Book chapter, 2021) ''Author note: TBC keep''
* [[Motivation and emotion/Book/2013/Intrinsic motivation|Intrinsic motivation]] (Book chapter, 2013)
* [[Motivation and emotion/Book/2025/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2025)
* [[w:Self-determination_theory|Self-determination theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Grubbs, J. B., Tóth-Király, I., Nagy, L., Koós, M., Kraus, S. W., Demetrovics, Z., Potenza, M. N., Ballester-Arnal, R., Batthyány, D., Bergeron, S., Billieux, J., Briken, P., Burkauskas, J., Cárdenas-López, G., Carvalho, J., Castro-Calvo, J., Chen, L., Strong, C., Ciocca, G., … Bőthe, B. (2025). Basic psychological needs satisfaction: An international examination of invariance across 22 languages and 32 countries. ''Motivation and Emotion'', ''49''(5), 552–574. https://doi.org/10.1007/s11031-025-10151-z

Montag, C., Lachmann, B., Herrlich, M., &amp; Zweig, K. (2019). Addictive features of social media/messenger platforms and freemium games against the background of psychological and economic theories. ''International Journal of Environmental Research and Public Health'', ''16''(14), 2612. https://doi.org/10.3390/ijerph16142612

Parent, N. (2023). Basic need satisfaction through social media engagement: A developmental framework for understanding adolescent social media use. ''Human Development'', ''67''(1), 1–17. https://doi.org/10.1159/000529449 

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''The American Psychologist'', ''55''(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryan, R. M., &amp; Deci, E. L. (2018). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. The Guildford Press.

Ryan, R. M., Deci, E. L., Vansteenkiste, M., &amp; Soenens, B. (2021). Building a science of motivated persons: Self-determination theory’s empirical approach to human experience and the regulation of behavior. ''Motivation Science'', ''7''(2), 97–110. https://doi.org/10.1037/mot0000194

Schivinski, B., Christodoulides, G., &amp; Dabrowski, D. (2016). Measuring consumers’ engagement with brand-related social-media content. ''Journal of Advertising Research'', ''56''(1), 64–80. https://doi.org/10.2501/JAR-2016-004

Trunfio, M., &amp; Rossi, S. (2021). Conceptualising and measuring social media engagement: A systematic literature review. ''Italian Journal of Marketing'', ''2021''(3), 267–292. https://doi.org/10.1007/s43039-021-00035-8

Vansteenkiste, M., &amp; Ryan, R. M. (2013). On psychological growth and vulnerability: Basic psychological need satisfaction and need frustration as a unifying principle. ''Journal of Psychotherapy Integration'', ''23''(3), 263–280. https://doi.org/10.1037/a0032359

Vansteenkiste, M., Ryan, R. M., &amp; Soenens, B. (2020). Basic psychological need theory: Advancements, critical themes, and future directions. ''Motivation and Emotion'', ''44''(1), 1–31. https://doi.org/10.1007/s11031-019-09818-1

West, M., Rice, S., &amp; Vella-Brodrick, D. (2023). Adolescent social media use: Cultivating and constraining competence. ''International Journal of Qualitative Studies on Health and Well-being'', ''18''(1), 2277623. https://doi.org/10.1080/17482631.2023.2277623 

West, M., Rice, S., &amp; Vella-Brodrick, D. (2024). Adolescent social media use through a self-determination theory lens: A systematic scoping review. ''International Journal of Environmental Research and Public Health'', ''21''(7), 862. https://doi.org/10.3390/ijerph21070862

}}
==External links==
* [https://selfdeterminationtheory.org/topics/application-basic-psychological-needs/ Basic psychological needs] (Center for Self-Determination Theory)
* [https://wearesocial.com/uk/blog/2025/10/digital-2026-global-overview-report/ Digital 2026 global overview report] (We Are Social)
* [https://www.ted.com/talks/margaret_gould_stewart_how_giant_websites_design_for_you_and_a_billion_others_too?referrer=playlist-how_did_the_internet_take_over_our_lives&amp;autoplay=true How giant websites design for you (and a billion others, too), Margaret Gould Stewart] (TED Talk, 2014) ''Author note: TBC keep''
* [https://selfdeterminationtheory.org/ Self-determination theory] (Center for Self-Determination Theory)
* [https://www.beyondblue.org.au/mental-health/Social-media Social media and mental health] (Beyond Blue)
* [https://www.youtube.com/watch?v=4TMPXK9tw5U The battle for your time: Exposing the costs of social media, Dino Ambrosi] (TEDx Laguna Blanca School, 2023)
* [https://www.youtube.com/watch?v=VGrcets0E6I&amp;t=2s Promoting motivation, health, and excellence, Ed Deci] (TEDx Flour City, 2012) ''Author note: TBC keep''
* [https://www.youtube.com/watch?v=m6fm1gt5YAM Edward Deci - Self-determination theory] (The Brainwaves Video Anthology, 2017) ''Author note: TBC keep''
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==Overview==
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[[File:Social media icons (rubin-Social-media-icons-2x).tiff|thumb|150px|'''Figure 1.''' Social media networks. ''(Author note: Alternatively use image of person using social medi''|right]]

'''Consider this scenario'''

You’re taking a quick study break and plan to make a cup of tea. While the kettle boils, you hear a ding and it is an Instagram notification showing that your friend has sent you a message. You then reply to that friend feeling briefly connected (see Figure 1). You spot a reel on psychology and self-determination theory and choose to watch, telling yourself it’s still related to your studies. But then the algorithm pulls you in. Psychology reels lead to study hacks, then lifestyle tips, then memes, then sheep jumping in fields. You come across a scrolling check-in video and quickly swipe down to keep scrolling. You stop noticing what you're choosing. Thirty minutes pass. Your tea is cold. You started in control, with purpose and feeling connected. Now it’s unclear what you were looking for and you feel less motivated to study. You didn’t mean to spend so long. Where did the time go?

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This chapter explores [[w:Self-determination_theory|self-determination theory]] (SDT) and how basic psychological needs (BPNs) according to [[Motivation and emotion/Book/2020/Basic psychological need theory|basic psychological need theory]] (BPNT) can explain patterns of social media engagement. Social media is becoming increasingly intertwined with (deeply &lt;u&gt;embedded in&lt;/u&gt;) everyday life, influencing how people communicate, relate &lt;u&gt;(to one another),&lt;/u&gt; and see the world and themselves/present themselves &lt;u&gt;--&gt; and at a fundamental level how people think, feel, and act/behave&lt;/u&gt; (&lt;u&gt;ref&lt;/u&gt;). 

* Pervasiveness of social media. Social media is deeply embedded in everyday life, influencing how people connect, communicate, and present themselves.
* The problem. While social media offers benefits like connection and self-expression, it is also linked to negative outcomes such as anxiety, low self-esteem, and poor well-being, particularly with excessive or unbalanced use.
* The need for psychological science. Understanding why people engage with social media and how it affects their mental health requires a psychological science perspective with relevant theories and evidence explaining the link between social media, motivation and well-being.
* Introduction to Self-Determination Theory (SDT). SDT is a well-established motivation theory that helps explain human behaviour by focusing on the satisfaction of three core psychological needs.
* Basic Psychological Need Theory (BPNT). A mini-theory of SDT, BPNT identifies three universal psychological needs: Autonomy (the need to feel in control), competence (the need to feel capable and effective), and relatedness (the need to feel socially connected and a sense of belonging) (Vansteenkiste et al., 2020). These BPNs are drivers of human behaviour and motivation.
* How social media interacts with these needs. Social media can support these needs. It can also frustrate them (e.g. comparison or social pressure)
* The role of design. Social media platforms are intentionally designed to engage users by targeting these psychological needs
* Why this matters. Applying SDT offers a deeper understanding of social media engagement and highlights ways to promote healthier, more intentional use that supports well-being.
* [[psychological needs]]
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'''Focus questions'''

* What is the relationship between social media, motivation and well-being?
* What psychological processes underpin social media design and patterns of engagement?
* What is self-determination theory?
* What are basic psychological needs according to self-determination theory?
* How do basic psychological needs explain patterns of social media engagement?
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== Social media, motivation and well-being ==
'''Focus question''': What is the relationship between social media, motivation and well-being?

Global prevalence of social media and integration into daily lives. The evolving landscape of digital technology including social media, also called social network sites (SNS), means virtual realms are increasingly integrated into everyday life and create new pathways for engagement and interaction. Excessive social media use poses significant concerns and consequences at individual and societal levels.

Distinguish between social media use and [[w:Problematic_social_media_use|problematic social media use]] which is associated with internet addiction disorder, a form of behavioural addiction (ref).

* Social media use is directly linked to screentime use as well as smartphone use. Statistics show social media is primarily accessed via smartphones (references). To find out more check out this chapter on [[Motivation and emotion/Book/2025/Mobile phone use motivation|mobile phone use motivation]].
* ''Author note: Expand on or link to issues of social media engagement and patterns of engagement. Focus on how social media impacts wellbeing and the psychological impacts of social media use.''
* There is a growing body of research examining the relationship between social media and well-being with motivation a key element linking drivers of social media engagement. Need for research using theoretical frameworks.
* Social relationships are at the heart of social media. These platforms were designed to help people communicate, stay in touch with friends and family, and share information. While their original purpose may have been to support social connection, social media has since evolved into a powerful tool with an attention-based based business model (Montag et al., 2019). ''Aurthor note check ref if 2019 0r 2021''
* Studies have shown social media can have both a positive and negative impact on well-being (Parent, 2023; West et al., 2024). Understanding what motivates people to use social media is critical to understanding how social media impacts well-being and what can be done to promote healthy social media use.
* [[Motivation]] is the energy for action (Ed Deci...)... quality of behavior not quantity of behavior

* The average person spends 145 minutes per day on social media (ref).
* Short-form videos are the most engaging content form on social media often being under 1 minute in length, capture attention due to bite-sized format (ref)
* Majority social media users access social media via mobile phones, increasing social medias accessibility and prevalence in everyday life.
* Demographic youth and young adult's social media usage skewed toward younger demographics with 
* &quot;Correlation between social media and depression for adolescents&quot; Time spent e.g. &quot;Adolescents who spend more than 3 hours a day have an increased risk of mental health struggles.&quot;
* Intrinsic vs extrinsic motivation --&gt; link to BPNs and intrinsic motivation
== The psychology behind social media design and patterns of engagement ==
[[File:Social media integration and central role in everyday life - media networks and connection.png|thumb|350x350px|'''Figure 2.''' Intergration and central role of social media in everyday life. Social media is increasingly becoming interwoven with everyday life, social connection and belongin]]'''Focus question:''' What psychological processes underpin social media design and patterns of engagement?

* Understanding the psychology behind social media design is important as it informs patterns of social media engagement and the psychological processes social media is designed to act on.
* ''Author note: Research more into psychological processes and possibly principles behind social media design and that are targeted when people use social media. While potentially not necessary, this section provides the opportunity to expand on the problem, how social media is closely, possibly intrinsically, linked to psychology and psychological science, briefly look at how other theories or aspects of motivation relate to social media, and lead into why self-determination theory and BPNT is a suitable (or the best/ideal) lens through which explain patterns of social media engagement.''

=== Social media design ===
* Social media is intentionally designed to capture one's/people's attention and hold it for as long as possible and includes addictive features (Montag et al., 2019).
* Also consider physiological and neurological processes that social media is designed to engage e.g. dopamine cycle
* Reward-seeking, social validation etc.
* Discuss theoretical and psychological underpinnings behind social media design and how it connects with psychological science, self-determination theory, BPNT, motivation and therefore why psychology can be used to understand engagement patterns. --&gt; ''potentially move above self-determination theory section''
* (see Figure 2)

=== Patterns of social media engagement ===

* Social media engagement is multi-dimensional and concerns affective, cognitive, and behaviour dimensions (Trunfio and Rossi, 2021).
* The Consumer Online Brand Related Activities Model (COBRA) is a conceptual tool used to classify and interpret social media engagement with brands and differentiates between three levels: consumption, contribution and creation (Schivinski et al., 2016). These levels of social media engagement encompass various actions including viewing, liking, sharing, commenting, and content creation (Trufino &amp; Rossi, 2021).
* Expand on problematic social media use compared to healthy patterns of social media engagement.
* ''Author note: Potentially split this section between sections: Social media, motivation and well-being AND basic psychological needs and social media engagement.''

== Self-determination theory ==
'''Focus question:''' What is self-determination theory?[[File:Self-Determination-Theory-Visual 1.png|thumb|'''Figure 3.''' Self-determination theory.|400x400px]][[Self-determination theory]] (SDT) is a key theory of [[w:Motivation|motivation]], playing a central role in [[w:Psychology|psychology]] and empirical studies on motivation (Ryan et al., 2021). SDT highlights human capacities and innate resources for behavioural self-regulation and personality development (Ryan et al., 2021).
* A key component of self-determination theory is the sub-theory basic psychological need theory (see Figure 3).
* SDT was first developed by Ryan and Deci and has since been widely used in research and evidence-based application across diverse fields such as health care, education, work, sport, technology use, psychotherapy, parenting, developmental and organisation psychology (Ryan et al., 2021).
* SDT is a well-established and robust theory, recognised for its universality/as a universal theory across cultures and domains (Ryan et al., 2021). Multi-dimensional wellbeing framework.
* SDT divides motivation into two categories: [[w:Intrinsic_motivation|intrinsic motivation]] and [[w:Extrinsic_motivation|extrinsic motivation]] (Ryan et al., 2021).
* Autonomous (intrinsic and extrinsic motivation) and controlled motivation.

* Possibly include figure of &quot;Self-determination theory's taxonomy of motivation&quot; (see Figure 4).
* Motivation is the and is related to activation and intention (Ryan &amp; Deci, 2000).
* SDT is comprised of six mini-theories: cognitive evaluation theory (CET), organismic integration theory (OIT), causality orientations theory (COT), basic psychological needs theory (BPNT), goal contents theory (GCT), and relationships motivation theory (RMT) (Ryan &amp; Deci, 2018). ''Aurthor note: Check ref 2017 or 2018.''
[[File:Amp-55-1-68-fig1a.gif|center|thumb|758x758px|'''Figure 4.''' Self-determination theory's taxonomy of motivation. ]]

== Basic psychological need theory ==
'''Focus question:''' What are basic psychological needs according to self-determination theory?

Basic psychological need theory (BPNT) is a mini theory/sub-theory within the SDT framework (Vansteenkiste et al., 2020). BPNT posits humans have three basic psychological needs: autonomy, competence, and relatedness (Ryan &amp; Deci, 2000) (see Figure 5). Psychological needs are critical resources underlying the inherent desires to engage with the environmental to advance personal growth, social development, self-organisation, and psychological well-being (Reeve, 2018; Vansteenkiste et al., 2020). According to BPNT BPNs are universal and must be satisfied for &lt;u&gt;(are crucial to)&lt;/u&gt; optimal well-being and performance, and if unsatisfied leads to negative consequences such as ill-being (Ryan et al., 2021). BPNT emphasises the importance of creating optimal circumstances to support peoples' basic psychological needs, with social and sociocultural contexts being a key/the main avenue for nurturing or thwarting needs (Vansteenkiste et al., 2020). BPNs drive human behaviour being intrinsically linked to motivation. 
According to BPNT, a need must meet nine conceptual and empirical criteria. The basic criteria dictate a need must be psychological, essential, inherent, distinct, and universal. The associated criteria state a need must also be pervasive, content-specific, directional, and explanatory (Vansteenkiste et al., 2020).[[File:Basic needs.png|thumb|'''Figure 5.''' Theory of basic psychological needs. &quot;'''Figure #'''. The three basic psychological needs proposed by self-determination theory are considered essential ingredients for intrinsic motivation and psychological well-being.&quot;]]
* Ryan and Deci (2000) define basic needs as energising states which when satisfied contribute toward health and well-being, however if unsatisfied lead to ill-being.
* BPNT initially
* &quot;Drivers of human action essential for psychological growth and activity&quot;
* Basic needs drive behaviour in order to satisfy/fulfill said needs.
* &quot;Basic psychological needs important motivational tools...&quot;
* BPNT relevant in social and non-social contexts.
* Universal psychological nutrients (Ryan &amp; Deci, 2018).
* &quot;Intrinsic motivation is fostered by conditions which support autonomy, competence, and relatedness (Lecture 4)&quot;
* BPNs cross-culturally (Grubbs et al., 2025)
=== Autonomy ===
Autonomy refers to the psychological need to experience willingness, volition, and self-organisation (Ryan et al., 2021; Vansteenkiste et al., 2020).

* &quot;...refers to the capacity to make informed and independent decisions.&quot;

* Autonomy is satisfied when actions, thoughts and feelings are self-endorsed and authentic creating the experience of integrity (Vansteenkiste et al., 2020).
* Autonomy is frustrated when conflict or pressure is felt that opposes one volition (Vansteenkiste et al., 2020).
* Autonomy support: their perspective, self-initiation, choice, explore, and rationale. 

=== Competence ===

* Competence is the psychological need to experience effectiveness and mastery (Vansteenkiste et al., 2020).
* &quot;...ability to do task effectively and efficiently.. improves task performance --&gt; task mastery. 
* Competence is satisfied when one is able to utilise and develop skills and expertise (Vansteenkiste et al., 2020). &quot;...engaging in activities on can make use of their skills and expertise.&quot;
* Competence is frustrated by experiences of failure, ineffectiveness, and helplessness (Vansteenkiste et al., 2020).
=== Relatedness ===

* Relatedness refers to is the psychological need to experience bonding, care, and warmth from others (Vansteenkiste et al., 2020).
* &quot;..need for connection, satisfied through bonding and care.&quot;
* Relatedness is closely tied to relationships and social connection and is satisfied by feeling connected to others (Vansteenkiste et al., 2020).
*Relatedness is frustrated by social exclusion, social alienation, and loneliness (Vansteenkiste et al., 2020).

== Basic psychological need satisfaction and frustration ==
* Expression of basic psychological needs varies across cultures due to the internalisation and integration of cultural values and behaviours facilitating need satisfaction (Ryan &amp; Deci, 2000).
* Need satisfaction is foundational for wellness and healthy functioning (Ryan et al., 2021).
* Need frustration is detrimental to well-being and diminishes integrity (Ryan et al., 2021).

=== Need satisfaction and need support ===
* SDT and BPNT are primarily concerned with fostering human growth and wellness through need satisfaction and need support (Vansteenkiste &amp; Ryan, 2013).
* The satisfaction of basic needs is facilitated by need supportive environments and promotes wellness, the development of inner resources, and resilience (Vansteenkiste &amp; Ryan, 2013).
* Satisfaction's role in motivation is to energise and sustain intrinsic motivation and internalisation (Ryan et al., 2021).

=== Need frustration and need thwarting ===
* Need frustration and need thwarting are associated with negative outcomes such as ill-being, self-control breakdowns, constricted functioning, externalising problems and increased defensive and immoral functioning (Vansteenkiste &amp; Ryan, 2013).
* &quot;...emotional response to opposition.&quot; 
* Need frustration and need thwarting are worse than the absence of need satisfaction and need support (Vansteenkiste &amp; Ryan, 2013).
* Needs frustration diminishes the energisation and sustenance of intrinsic motivation and internalisation, and elicits defiance (Ryan et al., 2021).
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- Autonomy

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+ Thwart
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== Basic psychological needs and social media engagement ==
'''Focus question:''' How do basic psychological needs explain patterns of social media engagement?

* BPNT can help explain patterns of social media engagement... (ref).
* Social media creates unique social contexts which
* Different factors of social media engagement can contribute to basic psychological need satisfaction and frustration including intra-personal, inter-personal, situational and environmental factors (West et al., 2024)
* The first scoping review examining the relationship between social media use and well-being in adolescents through a SDT lens found social media use can be both beneficial and detrimental to satisfying basic psychological needs of autonomy, competence and relatedness (West et al., 2024). Figure # represents how engagement with social media can satisfy BPNs supporting well-being, performance, and growth, however unfulfilled needs are not conducive for individuals to thrive.

*Table 1 shows examples of
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[[File:Social media use impact girls mental health plants thrive or wilt raw.png|left|thumb|350x350px|'''Figure 6.''' Impact of social media use on mental health and basic psychological needs. Plants represent thriving and fulfillment of needs vs frustration of needs ''([[ccorg:licenses/by-sa/4.0|Mercy Canning, CC BY-SA 4.0]], via Wikimedia Commons).'']]

=== Autonomy and social media engagement ===

* Social media supports autonomy by creating a ''(enabling one to experience a)'' sense of control, self-governance and personal agency (West et al., 2024).
* Social media has the potential to influence autonomy through identity development (West et al., 2024).
* Social media can challenge and threaten autonomy as choices and actions may be constrained or limited and individuals may be susceptible to other people's actions such as controlling behaviours (West et al., 2024).
* Expand on need satisfaction and support for autonomy
*Expand on need frustration and thwarting for autonomy
*Passive social media use is associated with lower autonomy and ill-being (ref)
*(see Figure 6)
=== Competence and social media engagement ===

* Sense of self can be positively or negatively impacted by social media (West et al., 2024). Areas related to sense of self that may be influenced by social media include self-esteem, confidence, self-concept, social competence... 
* Other areas impacted by social media include learning, digital competence (West et al., 2024). 
* The impact of social media on competence is consistent with other basic psychological needs in that social media may support or thwart competence (West et al., 2024)
* A study by West, Rice and Vell-Brodick (2023) found social media contributes to adolescent's experience of unique challenges regarding competence and identified two broad themes: cultivating competence and constraining competence.
* Expand on need satisfaction and support for competence
*Expand on need frustration and thwarting for competence
[[File:Social media positive &amp; negative effects on mental health &amp; well-being.png|thumb|350x350px|'''Figure 7.''' Description.]]

=== Relatedness and social media engagement ===

* Social media plays a key role in satisfying and frustrating the basic psychological need of relatedness (West et al., 2024). 
* Social media may be used to experience a sense of belonging and acceptance (West et al., 2024). 
* Social media enables parasocial relationships. Social media can support well-being when used for self-disclosure.

*Expand on need satisfaction and support for relatedness
*Expand on need frustration and thwarting for relatedness
*(see Figure 7)
*(see Table 1) organise and summarise info 
'''Table 1.''' 

Basic Psychological Needs Role in Motivation and Influence on Social Media Patterns of Engagement 
{| class=&quot;wikitable&quot;
!Basic psychological need
!Role in motivation
!Applied to social media engagement
|-
|[[w:Autonomy|'''Autonomy''']]
|
|
|-
|'''[[w:Competence|Competence]]'''
|
|
|-
|'''Relatedness'''
|Social connection

*

*
|
|}
''Author note: Either incorporate need satisfaction and frustration under related basic need level 1 sub-headings as two separate paragraphs (with or without level 2 sub-heading) OR create new major heading addressing need satisfaction and frustration. However, basic psychological needs explaining social media use naturally addresses satisfaction and frustration while explain how each basic need is related to social media. Consider structuring paragraphs under level 1 sub-heading by topic rather than needs satisfaction and frustration e.g. different types of competence social media influences such as sense of self and digital competence.''

''Author note:'' Potentially add case studies or examples of patterns of social media use explained by basic psychological needs for key age groups across the lifespan. Alternatively, case studies or examples in different environments/contexts.

* Case study on youth/adolescents
* Case study on university students
* Case study on adults or elderly adults


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== Promoting health and well-being through social media engagement ==
''Author note: Potentially add additional major heading and section focused on action orientated tips for healthy social media use that satisfies and supports basic needs and wellness.''

* Alt title: How to engage with social media to promote health and well-being through satisfaction of basic psychological needs

* Limiting social media use
* Use social media as an avenue for connection with friends and family while also prioritising connection with others in real-life face-to-face ways.
* Setting clear goals and motives for social media use
* Using social media as a creative outlet
* If you get struck
* '''Potential solutions?''' &quot;Limiting social media time, increasing media literacy and redesigning platforms might be keys to healthier digital interactions to counter this troubling trend. (Forbes).&quot;

==Conclusion==
* Social media use is a prevalent part of daily life for many people. Linked to increase in technology.
* SDT, specifically BPNT, helps explain social media patterns of engagement.

* Social media is deeply embedded in everyday life, offering opportunities to connect, express, and engage, but it also brings significant risks to mental health and well-being. 
* Psychological science, particularly SDT and BPNT, helps explain the complex relationship between social media, motivation, and well-being. According to BPNT, all humans have three basic psychological needs which are essential for psychological wellness and optimal performance: autonomy, competence, and relatedness. 
* Social media can support these needs by allowing users to make choices, build skills, and feel connected. However, it can also frustrate or thwart these needs, such as when users experience pressure, comparison, exclusion, or lack of control.
* The psychological processes behind social media design act on these basic needs and influence patterns of engagement that may feel rewarding but can become problematic over time. 
* Understanding underlying mechanisms of social media can help individuals and society make more informed, intentional decisions about how to engage with social media.
* Rather than framing social media as entirely good or bad, SDT provides a lens through which to understand its effects. 
* Supporting autonomy, competence, and relatedness is key to promoting healthy, self-determined social media use and enhancing overall well-being in a digital world.
* (see Figure 8)


''Author note: Address these points in conclusion''
* ''Answer to sub-title question based on psychological theory and research?''
* ''Answers to focus questions?''
* ''Practical, take-home messages?''
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* Message{{RoundBoxBottom}}
==See also==
* [[Motivation and emotion/Book/2025/Autonomy and intrinsic motivation in self-determination theory|Autonomy and intrinsic motivation in self-determination theory]] (Book chapter, 2025) 
* [[Motivation and emotion/Book/2020/Basic psychological need theory|Basic psychological need theory]] (Book chapter, 2020) ''Author note: TBC keep''
* [[Motivation and emotion/Book/2021/Frustration of basic psychological needs|Frustration of basic psychological needs]] (Book chapter, 2021) ''Author note: TBC keep''
* [[Motivation and emotion/Book/2013/Intrinsic motivation|Intrinsic motivation]] (Book chapter, 2013)
* [[Motivation and emotion/Book/2025/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2025)
* [[w:Self-determination_theory|Self-determination theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Grubbs, J. B., Tóth-Király, I., Nagy, L., Koós, M., Kraus, S. W., Demetrovics, Z., Potenza, M. N., Ballester-Arnal, R., Batthyány, D., Bergeron, S., Billieux, J., Briken, P., Burkauskas, J., Cárdenas-López, G., Carvalho, J., Castro-Calvo, J., Chen, L., Strong, C., Ciocca, G., … Bőthe, B. (2025). Basic psychological needs satisfaction: An international examination of invariance across 22 languages and 32 countries. ''Motivation and Emotion'', ''49''(5), 552–574. https://doi.org/10.1007/s11031-025-10151-z

Montag, C., Lachmann, B., Herrlich, M., &amp; Zweig, K. (2019). Addictive features of social media/messenger platforms and freemium games against the background of psychological and economic theories. ''International Journal of Environmental Research and Public Health'', ''16''(14), 2612. https://doi.org/10.3390/ijerph16142612

Parent, N. (2023). Basic need satisfaction through social media engagement: A developmental framework for understanding adolescent social media use. ''Human Development'', ''67''(1), 1–17. https://doi.org/10.1159/000529449 

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''The American Psychologist'', ''55''(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryan, R. M., &amp; Deci, E. L. (2018). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. The Guildford Press.

Ryan, R. M., Deci, E. L., Vansteenkiste, M., &amp; Soenens, B. (2021). Building a science of motivated persons: Self-determination theory’s empirical approach to human experience and the regulation of behavior. ''Motivation Science'', ''7''(2), 97–110. https://doi.org/10.1037/mot0000194

Schivinski, B., Christodoulides, G., &amp; Dabrowski, D. (2016). Measuring consumers’ engagement with brand-related social-media content. ''Journal of Advertising Research'', ''56''(1), 64–80. https://doi.org/10.2501/JAR-2016-004

Trunfio, M., &amp; Rossi, S. (2021). Conceptualising and measuring social media engagement: A systematic literature review. ''Italian Journal of Marketing'', ''2021''(3), 267–292. https://doi.org/10.1007/s43039-021-00035-8

Vansteenkiste, M., &amp; Ryan, R. M. (2013). On psychological growth and vulnerability: Basic psychological need satisfaction and need frustration as a unifying principle. ''Journal of Psychotherapy Integration'', ''23''(3), 263–280. https://doi.org/10.1037/a0032359

Vansteenkiste, M., Ryan, R. M., &amp; Soenens, B. (2020). Basic psychological need theory: Advancements, critical themes, and future directions. ''Motivation and Emotion'', ''44''(1), 1–31. https://doi.org/10.1007/s11031-019-09818-1

West, M., Rice, S., &amp; Vella-Brodrick, D. (2023). Adolescent social media use: Cultivating and constraining competence. ''International Journal of Qualitative Studies on Health and Well-being'', ''18''(1), 2277623. https://doi.org/10.1080/17482631.2023.2277623 

West, M., Rice, S., &amp; Vella-Brodrick, D. (2024). Adolescent social media use through a self-determination theory lens: A systematic scoping review. ''International Journal of Environmental Research and Public Health'', ''21''(7), 862. https://doi.org/10.3390/ijerph21070862

}}
==External links==
* [https://selfdeterminationtheory.org/topics/application-basic-psychological-needs/ Basic psychological needs] (Center for Self-Determination Theory)
* [https://wearesocial.com/uk/blog/2025/10/digital-2026-global-overview-report/ Digital 2026 global overview report] (We Are Social)
* [https://www.ted.com/talks/margaret_gould_stewart_how_giant_websites_design_for_you_and_a_billion_others_too?referrer=playlist-how_did_the_internet_take_over_our_lives&amp;autoplay=true How giant websites design for you (and a billion others, too), Margaret Gould Stewart] (TED Talk, 2014) ''Author note: TBC keep''
* [https://selfdeterminationtheory.org/ Self-determination theory] (Center for Self-Determination Theory)
* [https://www.beyondblue.org.au/mental-health/Social-media Social media and mental health] (Beyond Blue)
* [https://www.youtube.com/watch?v=4TMPXK9tw5U The battle for your time: Exposing the costs of social media, Dino Ambrosi] (TEDx Laguna Blanca School, 2023)
* [https://www.youtube.com/watch?v=VGrcets0E6I&amp;t=2s Promoting motivation, health, and excellence, Ed Deci] (TEDx Flour City, 2012) ''Author note: TBC keep''
* [https://www.youtube.com/watch?v=m6fm1gt5YAM Edward Deci - Self-determination theory] (The Brainwaves Video Anthology, 2017) ''Author note: TBC keep''
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Self-determination theory]]
[[Category:Motivation and emotion/Book/Social media]]</text>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Motivation and emotion book chapter */ Fix title</comment>
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      <text bytes="5648" sha1="hatddy5v3hkjcqbrzyasee3fg8knlly" xml:space="preserve">== Information about the author ==
Hi, my name is '''Mercy'''. I am studying a Bachelor of Science in Psychology at the [https://www.canberra.edu.au/ University of Canberra]. I enjoy the diversity within the field of psychology. I am particularly interested in the areas of psychotherapy, counselling, and mental health and wellbeing. 
[[File:Practicing mindfulness promotes creativity.png|thumb|300x300px|'''Figure 1.''' My key interests in psychology are mental health, psychotherapy, and counselling.  ]]

=== Professional interests ===
My current professional interests include:

* [[w:Mental_health|Mental health]] (see Figure 1)
* [[w:Psychotherapy|Psychotherapy]] and [[w:Counseling|counselling]]
* Social role valorisation
* Working with youth and young adults
* Understanding pain

=== Personal interests and passions ===
My personal interest and passions include:

* [[w:Art|Art]] and [[w:Photography|photography]]
* [[wikipedia:Ceramics|Ceramics]]
* Puzzles and boardgames
* Hiking
* Travel

== Professional profiles ==
Feel free to browse my [https://www.linkedin.com/in/mercy-canning-50619226b/ LinkedIn profile]. 

== Motivation and emotion book chapter ==
For my degree, I am undertaking a unit called [[motivation and emotion]]. I am currently contributing to the [[Motivation and emotion/Book|motivation and emotion book]] project by writing a book chapter for the [[Motivation and emotion/Book/2026|motivation and emotion book 2026]]. The [[Motivation and emotion/Book|motivation and emotion book]] project aims to help people understand and improve their motivational and emotional lives using psychological science. 

The book chapter I am writing is called [[Motivation and emotion/Book/2026/Basic psychological needs and social media use|Self-determination theory and social media use]] - How do basic psychological needs explain patterns of social media engagement?

== Social contributions ==

=== Wikiversity contributions ===

* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMotivational_dimensional_model_of_affect&amp;diff=2827645&amp;oldid=2761461 27 August 2026: Editied sub-heading casing.]
* [[Talk:Motivation and emotion/Book/2026/Excitement as an emotion#c-~2026-46800-82-20260827081600-Initial suggestions looking at excitment as an emotion|27 August 2026: Initial suggestions for developing book chapter on wikiversity.]]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMobile_phone_use_motivation&amp;diff=2827594&amp;oldid=2810725 27 August 2026: Editied book chapter sub-headings to use sub-heading 1 casing.]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FWork_motivation_and_self-determination_theory&amp;diff=2726464&amp;oldid=2676654 10 August 2025: Rewrote close-ended focus questions to be open-ended.]
* [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2020%2FBasic_psychological_need_theory&amp;diff=2739654&amp;oldid=2240585 20 August 2025: Commented on Basic psychological need theory book chapter and suggested sub-headings for the three basic needs with further information and examples underneath to improve clarity and page layout.]
* [[:File:Social media use impact girls mental health plants thrive or wilt.svg|18 October 2025: Created and uploaded an image to my book chapter titled 'Social media use impact girls mental health plants thrive or wilt.svg'.]]
* [[commons:File:Social_media_integration_and_central_role_in_everyday_life_-_media_networks_and_connection.png|20 October 2025: Created and uploaded an image to my book chapter titled 'Social media integration and central role in everyday life - media networks and connection.png'.]]
* [[commons:File:Social_media_use_impact_girls_mental_health_plants_thrive_or_wilt_raw.png|20 October 2025: Uploaded an updated image as a PNG file to my book chapter titled 'Social media use impact girls mental health plants thrive or wilt raw.png'.]]
* [[commons:File:Social_media_positive_&amp;_negative_effects_on_mental_health_&amp;_well-being.png|20 October 2025: Created and uploaded an image to my book titled 'Social media positive &amp; negative effects on mental health &amp; well-being.png'.]]

=== UCLearn contributions ===
*[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558?entry_id=806227 20 August 2026: Contributed to a UCLearn dsicussion titled 'Breathing Exercises and Relaxation Book Chapter' providing tips for development of the book chapter.]

* [https://uclearn.canberra.edu.au/courses/17386/discussion_topics/394550?entry_id=715257 20 August 2025: Commented on a discussion forum about what I want to learn about motivation and emotion.]
* [https://uclearn.canberra.edu.au/courses/17386/discussion_topics/408822 10 October 2025: Replied to a discussion post on UCLearn titled 'What is the hardest part to being motivated to achieve your academic goals?']
* [https://uclearn.canberra.edu.au/courses/17386/discussion_topics/405539 10 October 2025: Contributed to a UCLearn discussion post titled 'Music and Mood'.]
* [https://uclearn.canberra.edu.au/courses/17386/discussion_topics/409029 10 October 2025: Created a discussion post on UCLearn titled 'Why do you use (or avoid) social media? Exploring social media engagement and psychological needs'.]
* [https://uclearn.canberra.edu.au/courses/17386/discussion_topics/406646 11 October 2025: Contributed to discussion post on UCLearn titled 'Motivation for things you enjoy?']
* [https://uclearn.canberra.edu.au/courses/17386/discussion_topics/407763 11 October 2025: Replied to a discussion post about 'Balancing Self-Care and Being Productive'.]</text>
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  <page>
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      <text bytes="1174" sha1="gjox4b01lfeydqqh293d1f8nsdubzuz" xml:space="preserve">== '''About Me''' ==
Hello there, I'm '''Brontë'''!

I am an undergraduate student completing a [https://www.canberra.edu.au/course/780AA/4/2025 Bachelor of Science in Psychology] at the [[University of Canberra]]. My current project is being completed through the course [[Motivation and emotion/About|Motivation and emotion]], in which I'm writing a book chapter exploring the differences in motivation, through the lens of mindsets.
[[File:Filippini, La Lettura, Madame Bovary, 1881, Olio su tela, 60 x 68 cm.jpg|thumb|'''Figure 1.''' A painting of someone reading]]
Interests:

* Current read: [[wikipedia:Drive_Your_Plow_Over_the_Bones_of_the_Dead|Drive Your Plow Over the Bones of the Dead]] - [[wikipedia:Olga_Tokarczuk|Olga Tokarczuk]]
* [[wikipedia:Figure_skating|Figure Skating]]
* Youth mental health advocacy and volunteering 

== '''Book Chapter''' ==
I am developing a chapter exploring motivational mindsets! It will be titled [[Motivation and emotion/Book/2026/Prevention versus promotion mindset|Prevention versus Promotion Mindset: What are the motivational differences between prevention and promotion mindsets?]]

== '''Social Contributions''' ==

#
#
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      <text bytes="25091" sha1="dy40emnzclepii343j6jhsrdbxk2vu3" xml:space="preserve">{{/Banner}}
==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|U3280843}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|U3269672}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|U3280097}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|HawaSA}}
# [[/Athletic identity loss and returning to sport after injury/]] - How does injury related disruption to athletic identity affect motivation to return to sport? {{ME-By|Tammysaurus}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|Revial76}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|U3233213}}
# [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|U3175512}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|U3275984}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|Zozo}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Sienna33309}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|Vivekidid}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|LMM26}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|Kelp14}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|Sarah Hagan06}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistemic motivation and the need for cognitive closure influence our lives? {{ME-By|Ayat Al-kabai}}
# [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|U3260591}}
# [[/Expectancy-value theory of educational motivation/|Expectancy-value theory of educational motivation]] - What is expectancy-value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|StudentUC2026}}
# [[/Extended process model of emotion regulation/]] - What is the extended process model and how does it explain the regulation of emotions in different contexts? {{ME-By|TheHutt02}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|WonderfulKitten3}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|U3275899}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|U3216724}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|U3242837}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|Monuc9}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|U3286643}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|U3275940}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|U3297598}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|Reillyu3280706}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|Jshottt}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|U3283643}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? - User Name
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|Mim0502}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|U3280499}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|U3275909}}
# [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|U3261236}}
# [[/Motivating virtual teams/]] - How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarceration on Indigenous Australians?{{ME-By|U3183521}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|U3280743}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|Anne-Lyse Iran}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|BellaJohnson1}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|Gracelp}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|U3222012}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|U3284308}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|Jack4234}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|PsychstudentUniversity!}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|Bronte.H}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|U3203283}}
# [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|U3261207}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|U3274291}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|U3253363}}
# [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|Ella Kay244}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|U3246286}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|U3275908}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|GraceInMind}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|U3262868}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|U3236349}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|U3261376}}
# [[/Sleep deprivation, motivation, and academic performance/|Sleep deprivation, motivation, and academic performance]] - How does sleep deprivation affect motivation, attention, and academic performance in university students? {{ME-By|RileyRuckus}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|U3284302}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|U3281503}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|U3188047}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|Chloebateup}}
# [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does it affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|Ella234567}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|U3275775}}
# [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|J.M.A Watson}}
# [[/Windfall gain effect/]] - How does unexpected wealth influence behaviour and decision-making? {{ME-By|U3280122}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|SJPiper}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|U3282656}}
# [[/Adaptive versus maladaptive self-reflection/]] - When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|U3211150}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|Charlie.henderson1}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|JessJ117}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|LazPulch}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|Amirrorslens}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|U3269915}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|Amyuniversity}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|E3297976}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] - How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|U3068253}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|U3280159}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|U3228742}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|U3270398}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|U3243776}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians? {{ME-By|Lilfish215}}
# [[/Emotional expressivity/]] - What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|U3283812}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|Tofu05}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|U3239236}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] - What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}}
# [[/Emotion regulation ability and strategy/]] - How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|KB3250298}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy and jury decision-making/]] - How does empathy toward defendants and victims influence jurors' reasoning and verdict decisions? {{ME-By|U3254168}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|U3143751}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|Diaz Chas}}
# [[/Envy in the workplace/]] - What role does envy play in workplace behaviour? {{ME-By|Flickstar888}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|U3292769}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|ChillPsychGuy0607}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Growth mindset and psychological wellbeing/]] - How does a growth mindset influence psychological wellbeing? {{ME-By|Avj.06}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Immersive therapy for PTSD treatment/]] - How does it work and what are the effects? {{ME-By|StretchBeyond}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|Safiaah}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|U3330981}}
# [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3275992}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|U3246588}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|U3224236{{ME-By|
# [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|Katelyn Rod}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] - How do different mood states impact attention, memory, and problem solving? {{ME-By|U3283879}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|U3239251}}
# [[/Moral disgust and jury decision-making/]] - How does moral disgust influence jurors' judgments of guilt, blame, and punishment? {{ME-By|Yellowvines}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|Honeybelle11}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? - [[User:U3275873|u3275873]]
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|Mymunu}}
# [[/Perfectionism and athlete mental health/]] - How does perfectionism affect athlete mental health? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|U3243961}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|P U3270518}}
# [[/Psychedelic treatment of eating disorders/]] - How might psychedelic-assisted therapy influence psychological mechanisms involved in eating eating disorders? {{ME-By|Leilab23}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|User Name}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|User Name}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|U3282586}}
# [[/Romantic entertainment and love beliefs/]] - How do romantic entertainment influence beliefs and expectations about love and romantic relationships? {{ME-By|U3247927}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed?{{ME-By|U3279062}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|U3257744}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|Greg Philips}}
# [[/Self-blame and emotion/]] - How does self-blame influence emotional responses to negative events? {{ME-By|GU3281277}}
# [[/Self-disclosure and emotional intimacy/]] - How does self-disclosure foster emotional closeness in relationships? {{ME-By|U3283302}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|Pinkk47}}
# [[/Social connection and emotion regulation/]] - How do social relationships help regulate people's emotions? {{ME-By|U3284040}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|U3253354}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|ChelsSchofield}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|Safiaah}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? - U3284437
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteers' subjective wellbeing? {{ME-By|U3216851}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|U3239431}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|Mort006}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|U3286962}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|U3263365}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|U3306498}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|u3348724}}
# [[/Reinforcement sensitivity theory/]] - How does reinforcement sensitivity theory explain individual differences in motivation and emotion? - User Name
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|Karabi Tasneem}}
# [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|Med.011387}}

[[Category:Motivation and emotion/Book/2026]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|U3280843}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|U3269672}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|U3280097}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|HawaSA}}
# [[/Athletic identity loss and returning to sport after injury/]] - How does injury related disruption to athletic identity affect motivation to return to sport? {{ME-By|Tammysaurus}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|Revial76}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|U3233213}}
# [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|U3175512}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|U3275984}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|Zozo}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Sienna33309}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|Vivekidid}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|LMM26}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|Kelp14}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|Sarah Hagan06}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistemic motivation and the need for cognitive closure influence our lives? {{ME-By|Ayat Al-kabai}}
# [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|U3260591}}
# [[/Expectancy-value theory of educational motivation/|Expectancy-value theory of educational motivation]] - What is expectancy-value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|StudentUC2026}}
# [[/Extended process model of emotion regulation/]] - What is the extended process model and how does it explain the regulation of emotions in different contexts? {{ME-By|TheHutt02}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|WonderfulKitten3}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|U3275899}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|U3216724}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|U3242837}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|Monuc9}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|U3286643}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|U3275940}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|U3297598}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|Reillyu3280706}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|Jshottt}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|U3283643}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? - User Name
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|Mim0502}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|U3280499}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|U3275909}}
# [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|U3261236}}
# [[/Motivating virtual teams/]] - How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarceration on Indigenous Australians?{{ME-By|U3183521}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|U3280743}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|Anne-Lyse Iran}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|BellaJohnson1}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|Gracelp}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|U3222012}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|U3284308}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|Jack4234}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|PsychstudentUniversity!}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|Bronte.H}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|U3203283}}
# [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|U3261207}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|U3274291}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|U3253363}}
# [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|Ella Kay244}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|U3246286}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|U3275908}}
# [[/Basic psychological needs and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|GraceInMind}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|U3262868}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|U3236349}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|U3261376}}
# [[/Sleep deprivation, motivation, and academic performance/|Sleep deprivation, motivation, and academic performance]] - How does sleep deprivation affect motivation, attention, and academic performance in university students? {{ME-By|RileyRuckus}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|U3284302}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|U3281503}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|U3188047}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|Chloebateup}}
# [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does it affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|Ella234567}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|U3275775}}
# [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|J.M.A Watson}}
# [[/Windfall gain effect/]] - How does unexpected wealth influence behaviour and decision-making? {{ME-By|U3280122}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|SJPiper}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|U3282656}}
# [[/Adaptive versus maladaptive self-reflection/]] - When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|U3211150}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|Charlie.henderson1}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|JessJ117}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|LazPulch}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|Amirrorslens}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|U3269915}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|Amyuniversity}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|E3297976}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] - How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|U3068253}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|U3280159}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|U3228742}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|U3270398}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|U3243776}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians? {{ME-By|Lilfish215}}
# [[/Emotional expressivity/]] - What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|U3283812}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|Tofu05}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|U3239236}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] - What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}}
# [[/Emotion regulation ability and strategy/]] - How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|KB3250298}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy and jury decision-making/]] - How does empathy toward defendants and victims influence jurors' reasoning and verdict decisions? {{ME-By|U3254168}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|U3143751}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|Diaz Chas}}
# [[/Envy in the workplace/]] - What role does envy play in workplace behaviour? {{ME-By|Flickstar888}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|U3292769}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|ChillPsychGuy0607}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Growth mindset and psychological wellbeing/]] - How does a growth mindset influence psychological wellbeing? {{ME-By|Avj.06}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Immersive therapy for PTSD treatment/]] - How does it work and what are the effects? {{ME-By|StretchBeyond}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|Safiaah}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|U3330981}}
# [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3275992}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|U3246588}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|U3224236{{ME-By|
# [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|Katelyn Rod}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] - How do different mood states impact attention, memory, and problem solving? {{ME-By|U3283879}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|U3239251}}
# [[/Moral disgust and jury decision-making/]] - How does moral disgust influence jurors' judgments of guilt, blame, and punishment? {{ME-By|Yellowvines}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|Honeybelle11}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? - [[User:U3275873|u3275873]]
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|Mymunu}}
# [[/Perfectionism and athlete mental health/]] - How does perfectionism affect athlete mental health? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|U3243961}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|P U3270518}}
# [[/Psychedelic treatment of eating disorders/]] - How might psychedelic-assisted therapy influence psychological mechanisms involved in eating eating disorders? {{ME-By|Leilab23}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|User Name}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|User Name}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|U3282586}}
# [[/Romantic entertainment and love beliefs/]] - How do romantic entertainment influence beliefs and expectations about love and romantic relationships? {{ME-By|U3247927}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed?{{ME-By|U3279062}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|U3257744}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|Greg Philips}}
# [[/Self-blame and emotion/]] - How does self-blame influence emotional responses to negative events? {{ME-By|GU3281277}}
# [[/Self-disclosure and emotional intimacy/]] - How does self-disclosure foster emotional closeness in relationships? {{ME-By|U3283302}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|Pinkk47}}
# [[/Social connection and emotion regulation/]] - How do social relationships help regulate people's emotions? {{ME-By|U3284040}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|U3253354}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|ChelsSchofield}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|Safiaah}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? - U3284437
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteers' subjective wellbeing? {{ME-By|U3216851}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|U3239431}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|Mort006}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|U3286962}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|U3263365}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|U3306498}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|u3348724}}
# [[/Reinforcement sensitivity theory/]] - How does reinforcement sensitivity theory explain individual differences in motivation and emotion? - User Name
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
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      <text bytes="41007" sha1="9iv4hpo45bhgk2u45a883urbs9l88ht" xml:space="preserve">{{title|Grit and academic achievement &lt;br&gt;What role does grit play in academic achievement and can it be fostered in future students?}}

== &lt;big&gt;Overview&lt;/big&gt; ==
{{robelbox|theme=7|title=Case study: Nikil|iconwidth=55px|icon=Search-icon-white-background.png}}

[[File:Case_study_magnifying_glass.png|left|340px|Looking closer at the case study! ]]

Nikil is a 21-year-old medical student in India who is a high-achieving student. He is the first in his family to attend a university. His family emphasizes that his success will bring honor and pride to his parents and community. He feels a great need to excel to repay his parents’ sacrifices and make use of his opportunity to gain higher socio-economic status. 

He often studies late into the night and skips meals, not because becoming a doctor is something he enjoys, but because he is afraid of disappointment and failure. He often ruminates heavily on his mistakes, compares his grades to others and has anxiety before exams. He often feels he must excel to be accepted. 

He began to find it hard to maintain a 97% average which he had throughout high school into university. Nikhil's current lifestyle left him feeling empty and fatigued, with decreased interest and creativity. Additionally, he overcriticized himself for not being perfect and it affected how he felt about himself. 

As he talked with his parents about how he was feeling, they helped him reflect on why he wanted to become a doctor, which helped him realise that this was exactly what he wanted to pursue. During his hospital rotation, he realised that his ability to empathise, think logically, and make a lasting impact on his client gave him immense satisfaction. He realised this is truly his career path. Once he realised this,  his studies became much more enjoyable; he was able to persist, committed, and develop healthy coping strategies, ultimately helping him to graduate successfully.

This case study shows Nikil is someone who has grit, particularly through his persistence and effort. However, support from his family and his own personal interest in becoming a doctor helped reinvigorate his passion and commitment to pursuing this career. {{RoundBoxBottom}}

Students around the world are faced with new challenges, setbacks, dissapointments{{sp}} and difficulties everyday. These setbacks can be especially challenging during the early educational years, when career sucess{{sp}} is predicated on academic performance. A recent example of a setback faced by students everywhere is the [[COVID-19|Coronavirus (Covid-19) Pandemic]] and the emergence of [[wikipedia:Artificial_intelligence_in_education|AI learning tools.]] This caused students, teachers and scientists around the globe to persevere against systemic-level disruptions by adaping{{sp}} and developing innovative ways to connect, learn, improve and retain knowledge in order to facillitate{{sp}} academic achievement. Some students found these challenges especially difficult leading them to disengage and drop out, but others like Nikhil (who we will talk about) persisted and eventually found success. This begs the question: is academic achievement based solely on your [[Intelligence Quotient|IQ]] score, talent, [[Psychological resilience|resilience]] or perhaps you need some good old, hard work? This exact question led author and psychologist [https://angeladuckworth.com/ Angela Duckworth] to popularise the term &quot;[[wikipedia:Grit_(personality_trait)|grit]]&quot;, suggesting persistence interest and sustained effort paved the way for success as opposed to talent or skills alone.

{{RoundBoxTop|theme=15}}
'''Focus questions:''' 
* What is grit?
* How does grit influence academic achievement?
* What psychological theories explain grit?
* What does the empirical evidence say about grit? Any critiques?
* Can grit be fostered and if so, how can it be applied?

'''Reflection:'''

'''Think of a time when you faced a major academic or life challenge. What factors or setbacks helped or hindered your ability to continue during the challenging situation?'''
{{RoundBoxBottom}}

Throughout the chapter, refering{{sp}} back to this case study of Nikil below to help consolidate your learning.

{{RoundBoxTop|theme=1}}
[[File:Crystal Clear app ktip.svg|left|20px|]]  
'''Key concepts:'''
*Grit
*Academic Achievement
* Persistence
* Interest
*Intelligence
*Consciousness
*Self-regulation
{{RoundBoxBottom}}

== &lt;big&gt;Grit&lt;/big&gt; ==
[[File:Marathon runner.png|thumb|289x289px|A middle-aged man showing his perseverance and passion towards achieving his goal of winning a marathon.]]

Grit is often regarded as a conceptual topic without a definitive definition or measure, in order to quantify, measure and understand what laypeople and scientists regard grit to be lets look at a few sources which ain to define grit.

=== &lt;u&gt;&lt;big&gt;Defining grit&lt;/big&gt;&lt;/u&gt; ===
The word &quot;grit&quot; originates from the German word ''&quot;[https://www.collinsdictionary.com/dictionary/german-english/griess Griess],&quot;'' which according to the [[mwod:grit|Merriam-Webster dictionary]], means &quot;grit&quot;.

Furthermore the dictionary defintion for a person with grit is explained as two qualities: firmness of mind or spirit and unyielding courage in the face of hardship or danger. Otherwise, as a person with courage and is adapt to grinding, setbacks, or hardships through the firmness of their mind. 

Although this is the dictionary definition, the psychological definition is slightly different. [[wikipedia:Angela_Duckworth|Angela Lee Duckworth]] and colleagues popularised &quot;grit,&quot; to be simply defined as a personality trait of an individual who [https://dictionary.cambridge.org/dictionary/english/persevere\ perseveres] and has a [https://dictionary.cambridge.org/dictionary/english/passion passion] for their long-term [[Goal setting|goals]] (Lam &amp; Zhou, 2019; Lee, Kim &amp; Shin, 2025; Tugabirwe &amp; Rukundo, 2024; Jiang et al., 2019 )

[[File:Girl climbing mountain.png|thumb|256x256px|A girl through passion and perseverance, is climbing difficult and challenging mountain to achieve her long-term goal of climbing to the summit{{ic|how does this relate to academic achievement?}}]]

&lt;blockquote&gt;
''&quot;Grit is passion and perseverance for long-term goals.''

''One way to think about grit is to consider what grit isn’t. Grit isn’t talent.'' 

''Grit isn’t luck. Grit isn’t how intensely, for the moment, you want something.''

''Instead, grit is about having what some researchers call an &quot;ultimate concern”–a goal you care about so much that it organizes and gives meaning to almost everything you do. And grit is holding steadfast to that goal. Even when you fall down. Even when you screw up. Even when progress toward that goal is halting or slow.''

''Talent and luck matter to success. But talent and luck are no guarantee of grit. And in the very long run, I think grit may matter as least as much, if not more.&quot;''  

[https://angeladuckworth.com/qa/#faq-125 ''━'' ''Angela Duckworth'']&lt;/blockquote&gt;

The definition used by the American Psychological Association is,

&lt;blockquote&gt;''&quot;n. a personality trait characterized by perseverance and passion for achieving long-term goals. Grit entails working strenuously to overcome challenges and maintaining effort and interest over time despite failures, adversities, and plateaus in progress. Recent studies suggest this trait may be more relevant than intelligence in determining a person’s high achievement. For example, grit may be particularly important to accomplishing an especially complex task when there is a strong temptation to give up altogether.&quot;''

[https://dictionary.apa.org/grit ''━'' ''APA 2018'']&lt;/blockquote&gt;

=== &lt;u&gt;&lt;big&gt;Characterising grit&lt;/big&gt;&lt;/u&gt; ===
[[File:Elderly + award.png|thumb|283x283px|A person, despite age {{g}} achieved an award through their grit.]]

Grit is characterised as a hierarchical, non-cognitive dispositional or personality trait which means, an enduring characteristic and behaviours employed by the individual to uniquely adjust to life across different situations (APA, 2018; Barbouta, Barbouta &amp; Kotrotsiou, 2020; Clark, Dorio, Eldridge, Malecki &amp; Demaray, 2020; Clark &amp; Malecki, 2019; He et al., 2021; Helal &amp; Hassan 2025; Lam &amp; Zhou, 2019; Lee &amp; Sohn, 2017; Singh &amp; Chukkali, 2021).   

A gritty individual works tirelessly towards challenges, maintains effort, interest and stamina over extended periods despite failures, advesities and plateus (He et al., 2021; Lee, Kim &amp; Shin, 2025; Tugabirwe &amp; Rukundo, 2024). Grit is a malleable construct which can be learned and developed through coping skills and self-cultivation, social support, training, experiences and practicing endurance similar to how you would develop your muscles in the gym (Duckworth et al., 2007; Harpaz, Vaizman &amp; Yaffe, 2024; Singh &amp; Chukkali, 2021; Jiang et al., 2019 ). Specifically, Grit encompasses goal-directness, persistence, passion and commitment.  

According to a 2020 study there is a moderate positive correlation between grit and older participants (Barbouta, Barbouta &amp; Kotrotsiou, 2020). Perhaps, due to the increase time to experience different situations and circumstances could be accredited to why older people tend to have higher grit levels. This could also be a contributing reason as to why postgraduate students in the same study had a higher overall mean in academic score of 78.2 (SD of 10.0) compared to the undergraduate students (overall mean of 72.9, SD of 9.6) (Barbouta, Barbouta &amp; Kotrotsiou, 2020).  

==== &lt;big&gt;2 Facet of grit&lt;/big&gt; ====
[[File:Mollusk, shells and fossils prize winner.png|thumb|224x224px|A fossil hunter holding their winning ammonite fossil earned through his lifelong passion and commitment to finding, and refurbishing pieces of fossil history.]]Perseverance of effort (PE) is the ability to sustain energy and towards long term goals involving high mental stamina by overcoming adversities and distractions through sustained effort to accomplish a goal (Clark &amp; Malecki, 2019; Lee, Kim &amp; Shin, 2025; Jiang et al., 2019 ; Wolters &amp; Hussain, 2015).

Oftentimes, PE is compared to qualities of [[wikipedia:Conscientiousness|conscientiousness]] from the big five personality theory because of the overlapping themes of being responsible, diligent and goal-oriented (O’Connor &amp; Paunonen, 2007).

Consistency of interest (CI) is ability to adhere and maintain focus to goals which the idividual is passionate and accredits meaning and purpose towards for long periods to sustain their ultimate final long term goal( Clark &amp; Malecki, 2019; Lee, Kim &amp; Shin, 2025; Jiang et al., 2019 ). For example, an ultimate final goal could be completing an undergraduate degree with a high distinction.

=== &lt;big&gt;&lt;u&gt;Measures of grit&lt;/u&gt;&lt;/big&gt; ===
There are multiple scales which have been used to quanitify grit. These incluse the Original Grit scale, Short Grit scale, Triarchic model of grit scale, Multi-dimensional grit scale and the Academic grit scale.

===== Original grit scale (Grit-O) =====
[[File:Psychological Survey.png|thumb|292x292px|Psychological self-report survey measure ]]

As the name suggests this is the original first developed grit scale by Duckworth and collegues, asking demographic questions, the individual's level of academic performance satisfaction compared to their actual academic score (Barbouta, Barbouta &amp; Kotrotsiou, 2020). Nowadays this scale is commonly used.

===== Short grit scale (Grit-S) =====
The Short Grit Scale is an improved, model fit and overall a better measurement tool inspired by the Grit-O. This is currently the most used grit scale in wester and individualist countries. It is a twelve item  two factor hierachical scale maximized by the particular items which were most correlated in accordance with factor analysis.

===== Triarchic model of grit scale =====
The triachic model of grit by Datu and collegues in 2017 expanded upon the 2 factor hierachical structure and introduced the facet of situational adaptability, which gain acclaim.

===== Multi-dimensional scale of grit =====
With the growing need for cultural inclusion and generalisability with research the Multi-Dimensional Scale of Grit (MDSG) was designed by Singh and Chukkali in 2021 to allow the concept of grit to traverse the global scale and be more accessible to non-western and individualistic communities (Singh &amp; Chukkali, 2021). This 12 item Likert scale measure, particularly designed to be applicable to collectivist, religious and eastern countries where through religious and ancient literature have identifies similar elements such as “gumption&quot; and “titiksha” from Hindu holy books; yang in Taoist principles; “dukkha” from Buddhist literature and the Islamic perspectives’ emphasis on strength of character, constancy, steadfastness, work devotion determination (Singh &amp; Chukkali, 2021). Overall the scale tends to have good internal consistency, reliability, criterion and divergent validity, but more studies need to conducted to corroborate its generalizability with the wider community due to the gender, age and university humanity and social science focus (Singh &amp; Chukkali, 2021).

==== Academic grit scale (AGS) ====
Finally, the concept of domain-specific grit was explore through the academic grit scale. This scale has 10 items focusing on determination, resilience, and focus in the pursuit of challenging long-term goals within education (Clark &amp; Malecki, 2019).

[[File:Glossary icon.jpg|centre|93x93px|'''&lt;u&gt;Glossary&lt;/u&gt;''' ]]

'''Table 1.''' The 2 x 7 on the definition and theory adjasent to grit x key theoretical terms.
{| class=&quot;wikitable sortable&quot;
|+&lt;big&gt;Key terms often associated with grit:&lt;/big&gt;
!Key term
!Definition
!Theory
|-
|[https://dictionary.apa.org/lockes-theory-of-goal-setting Long-term goal oriented:]
|

* ''a'' ''person' who focused on the process of establishing specific, time-based behavior targets that are measurable, achievable, and realistic. Goal setting is effective only if the individual's are aware of what needs to be accomplished and accept the goal themselves  and believe in their attainability.''
|According to Locke theory of goal setting, specifies specific, challenging, sub-goals are better for performance to regulate energy expenditure, improve persistence, skill development with timely feedback on progression are pivotal to attaining long term goals, such as academic achievement.
Also in accordance with self-determination theory a individual is moer{{sp}} likely to achieve their long term goals if he motivation behind their drive is intrinsic to satidy their psychological needs for autonomy, competence and relatedness (Deci &amp; Ryan, 2000).
Furthermore, expectancy-value theory suggests a persons effort, persistence and choices are determined by the person self-efficacy to attain goal (Deci &amp; Ryan, 2000). 

This allows the individual to foster a growth mindset where their belief in their abilities help them to self-regulate, make informed choices and dedicate themselves to their goal (Deci &amp; Ryan, 2000).
|-
|[https://dictionary.apa.org/passion Passionate]:
|
*''an intense, driving, or overwhelming feeling or conviction. Passion is often contrasted with emotion, in that passion affects a person unwillingly.''
*''a strong enthusiasm for or devotion to an activity, object, concept, or the like.''   
|According to self-determination theory, there are two types of passion: harmonious which is the healthy form compared to the maladaptive form called obsessive (Deci &amp; Ryan, 2000). 
Harmonius{{sp}} passion allows for flexibility and immersive enagement{{sp}}  whereas obsessive passion can be rigid and impersonal Deci &amp; Ryan, 2000).
|-
|[https://dictionary.apa.org/persistence Perseverance/Persistence:] 
[https://dictionary.apa.org/effortfulness Effortfulness]
|
* ''Continuance or repetition of a particular behavior, process, or activity despite cessation of the initiating stimulus.''
* ''The quality or state of maintaining a course of action or keeping at a task and finishing it despite the obstacles (such as opposition or discouragement) or the effort involved.''
* ''exertion that demands attentional and other cognitive resources: a feature of many psychological tasks that can be judged reliably by participants.''
|According to Cloninger’s seven factor psychobiological model of personality expand further on persistence as the tendency to continue a task or activity regardless of frustration, dissatisfaction, or fatigue.
According to SDT the psychological needs to be competence, related and sutonomous can increase the volition of the peron to strive in their activities (Deci &amp; Ryan, 2000).
|-
|[https://psycnet.apa.org/doiLanding?doi=10.1037/edu0000699 Academic achievement]
|
* ''any identifiable success in the areas of scholarship or disciplined study.''
* ''in educational psychology, a level of proficiency in scholastic work in general or in a specific skill, such as arithmetic or reading. Evidence of future academic achievement is usually based on the results of standardized ability tests and assessment of performance by a teacher or other supervisor.''
|Achievement goal theory talks about there are 4 major categories of goals: approach or avoidant mastery of skills and understanding and approach or avoidant performance or competive{{sPP and peer validation based goals. the avoidant types of goals tend to have negative affect, drive and self-worth outcomes (Harwood &amp; Thrower, 2020).
|-
|[https://dictionary.apa.org/interest Interest:]
|
* ''an attitude characterized by a need or desire to give selective attention to something that is significant to the individual, such as an activity, goal, or research area.''
|
|-
|[https://dictionary.apa.org/resilience Resilience:]
|
*''the process and outcome of successfully adapting to difficult or challenging life experiences, especially through mental, emotional, and behavioral flexibility and adjustment to external and internal demands.''
* ''A number of factors contribute to how well people adapt to adversities, predominant among them (a) the ways in which individuals view and engage with the world, (b) the availability and quality of social resources, and (c) specific coping strategies. Psychological research demonstrates that the resources and skills associated with more positive adaptation (i.e., greater resilience) can be cultivated and practiced.''
|
|-
|[https://dictionary.apa.org/intelligence Intelligence]
|
* ''the ability to derive information, learn from experience, adapt to the environment, understand, and correctly utilize thought and reason.'' 
|
|}

== Grit &amp; academic achievement: ==
The association between grit and academic achievement using the measure discussed earlier try to empircally understand and predict success in academic settings.

==== &lt;big&gt;Foundational studies&lt;/big&gt; ====
Foundational studies by Duckworth and colleagues tried to predict success through academic engagement, self-efficacy, educational graduation, retention rates and Grade point averages over time (Wen et al., 2019 cited from Duckworth et al., 2007). Throughout the years two types of grit was identified.

===== General Grit =====
* An overarching dispositional trait where attitudes and behaviors stay consistent across different contexts (Clark &amp; Malecki, 2019).

===== Domain-specific Grit =====
* Grit-based attitudes and behaviors are only displayed when in a certain contexts or condition (Clark &amp; Malecki, 2019).
* For example, Academic grit is when a person shows grit during educational pursuits.

Overall domain-specific grit has shown a better predictor of academic achievement than general grit. (Clark &amp; Malecki, 2019).

==== &lt;big&gt;Recent studies&lt;/big&gt; ====
Although foundational studies highlighted a high correlation between grit and academic achievement, especially amongst primary to college students, the current data suggest differently.{{f}}

[[File:Contradicting research data plots.png|thumb|240x240px|Contradicting research data plots from different research articles]]

A 2019 meta-analysis of suggested the correlation between grit and academic achievement was small but significant with the mean coefficient of 0.16 (95% CI: 0.14, 0.18 Z=15.95, p &lt; .001) and mean average correlation of overall grit is 0.17 (Lam &amp; Zhou, 2019). Similarly a more recent 2022 meta-analysis suggested the overall grit level to academic achievement was weak to moderate (weighted correlation = 0.19). They also found, the effect size is similar to conscientiousness (r = .19) and subjective well-being (r = .16). The total estimated average weighted average of grit overall was 0.24 (95% CI: 0.213, 0.261) (Lam &amp; Zhou, 2019).

In accordance with longitudinal studies, they seem to support a reciprocal relationship of PE and academic achievement being mutually reinforcing to each other (Wen et al., 2019). This means PE predicts achievement and prior achievement contributes to PE’s development. Overall, PE has been discovered to be a better predictor for academic success than CI especially in standardised measures like the gpa and similar school grades systems (Lam &amp; Zhou, 2022). Many studies consistently show PE has higher correlations and reforcing relationship (r=0.21; R&lt;sup&gt;2&lt;/sup&gt;=2.3%; r= 0.508; r=0.425) to academic success whereas, CI has a weaker predictive and correlation to academic success (Clark, Dorio, Eldridge, Malecki &amp; Demaray, 2020; Clark &amp; Malecki, 2019; Lam &amp; Zhou, 2022; Jiang et al., 2019 ).

Also, a two-factor hierarchical structure in the AGS scale validation studies with samples from high-achieving adolescents, the national spelling bee child and adolescent finalists, Ivy League undergraduates, and West Point cadets showed to have strong internal consistency, test-retest reliability, consensual validity and high predictive validity (Clark &amp; Malecki, 2019). There is also evidence to suggest grit is a better predictor of academic achievement in kindergarden to year 12 students (r=0.17) comparatively to university students (r=0.14)(Lam &amp; Zhou, 2019; Lam &amp; Zhou, 2022).

==== &lt;big&gt;Critiques:&lt;/big&gt; ====
{| class=&quot;wikitable&quot;
|+Table 2. The 2 x 16 on the Strengths and Weaknesses of Grit x Theory Efficacy
!
!Strengths
!Limitations
!Sources
|-
|'''Scope'''
|Domain-specific
|Overlaps with Consciousness
|Clark, &amp; Malecki, 2019; Jiang, et al., 2019). 
----(Clark, &amp; Malecki, 2019; Lam &amp; Zhao, 2022; Martoyo &amp; Lindawati, 2023; Rimfeld, Kovas, Dale &amp; Plomin, 2016).
|-
|'''Predictive power'''
|Validity (incremental, can build upon itself)
|Weak Consistency of Interest
|(Clark, &amp; Malecki, 2019; Harpaz, Vaizman &amp; Yaffe, 2024). 
----(Abubakar, et al., 2021; Jiang, et al., 2019; Lam &amp; Zhao 2022)
|-
|'''New perspective'''
|Non cognitive focus (included cultural backgrounds)
|Method issues with reliability and correlation
|(Jiang, et al., 2019; Singh &amp; Chukkali, 2021)
----(Clark, &amp; Malecki, 2019; Clark, Dorio, Eldridge, Malecki, &amp; Demaray, 2020; Jiang, et al., 2019; Lam &amp; Zhou, 2019; Tannoubi et al., 2023).
|}

All of these contradictions and critiques makes it difficult to determine whether grit is the sole influencer of academic success as once believed.

== Alternative academic success influences ==
As an alternative to thinking grit is sole contributing factors to academic success, let us, think more broadly. 

===== Cognitive Ability (IQ) =====
Grit is distinct from cognitive ability/IQ. Grit has been shown{{f}} to outperform IQ as a predictor of academic achievement in samples of high-achieving students.

Although grit is a better predictor of academic success in high achieving samples; student samples of lower end of the IQ spectrum, from poor socioeconomic status has suggested different results (Lee &amp; Sohn, 2017; Segal &amp; Kalfon-Hakhmigari, 2025; Tugabirwe &amp; Rukundo, 2024). In a 2021 study looking at disadvantaged rural students, grit had an insignificant effect on school achievement for those with an IQ of 0.85 (Jiang, et al., 2019).

A 2025 study suggested, grit could moderately influence cognitive abilities towards high educational achievement. It said stem high school student with high levels of cognitive abilities were statistically significant (β = 0.14, SE = 0.03, t(640) = 5.08, p &lt; 0.001, 95% CI [0.09, 0.19]) suggesting 34.58% of variance in STEM (physics, computer science, achievement interaction between cognition and grit to strongly predict highschool success (Segal &amp; Kalfon-Hakhmigari, 2025){{rewrite}}. Surprisingly, this study suggested a greater association between CI (r = 0.10 to 0.15 (p &lt; 0.05) as opposed to PE (r = 0.08, p = 0.046) as previous studies suggested helping them to bridge gaps which would otherwise affect performance (Segal &amp; Kalfon-Hakhmigari, 2025). Another study looking at Indonesian students and GPA also found similar results (1.4% - 6.3%) and grit scores seemed to increase as they progressed through school from  first to second to third year  (Martoyo &amp; Lindawati, 2023). As well as, students with extremely high levels of cognition showed slightly lower levels of grit compared to student with high cognition (Segal &amp; Kalfon-Hakhmigari, 2025){{g}}.

===== Deliberate practice =====
Deliberate practice ($r = .19, p &lt; .01$) and conscientiousness ($r = .21, p &lt; .01$) in korean college students correlated positively with GPA, this highlighted the grit did not have a direct effect but mediatory one to achievement in academics (Lee &amp; Sohn, 2017).

===== Self-regulated learning =====
Self-regulated learning suggests a consistent association of PE with self-efficacy, cognitive, metacognitive, motivational, time and study environment management strategies, and procrastination and CI with study environment management strategies, and procrastination, with only PE showing any improvement to academic achievement through regression analysis (Wolters &amp; Hussain, 2015).

===== Research motivation =====
Research motivation based on their intelligence (r = 0.72, p &lt; 0.01) in postgraduate student was correlated with grit (r = 0.27, p &lt; 0.01) and academic achievement (r = 0.26, p &lt; 0.01). Although research motivation was the bigger predictor with an r value of 0.84 (p &lt; 0.01), making it a critical mediatory pathway.

===== Coping and self-cultivation skills =====
Academic grit and coping strategies which included self-efficacy, autonomic Help-Seeking Orientation and self-cultivation or personal growth characteristics in Anglo-Saxon (0.37) participant, explained 24-26% of variance (Harpaz, Vaizman &amp; Yaffe, 2024).

===== Social Support =====
Social support plays a critical role by enhancing correlation between grit and academic achievement. The first study did a regression analysis on korean student and receiving parental support (r= 0.267, p&lt; 0.01), friend support (r= 0.140, p&lt; 0.01), teacher support (r= 0.197, p&lt; 0.01), and grit (r= 0.367, p&lt; 0.01), as being correlated with academics (Lee, Kim &amp; Shin, 2025).

== Conclusion ==
Grit's relationship with academic achievement has evolved over time. Foundational studies identified two types of grit: general grit, a consistent trait across contexts, and domain-specific grit, which manifests in specific areas like academics. Although early research linked grit strongly to academic success among students, recent meta-analyses have shown only a small correlation, with prior academic performance being a better predictor than grit. Critically, grit overlaps with cognitive ability. For disadvantaged students, grit has shown limited impact. Moreover, factors such as cognitive ability, deliberate practice, self-regulated learning, research motivation, and social support emerge as significant influences on academic achievement, indicating that grit alone may not be the sole determinant of success. 

Gaps within the research are the subjective of causality and the mechanisms for grit as a concept has led to various methodological limitations and reduce the reliance on self-reports which are subject to a multitude of biases. The overall data lack generalisability at time due to the hyper restricted populations of the sample derived for the study, with studies focusing on western and students with high cognitive abilities. Other factors like burnout, career aspirations, non-educational responsibility, cognitive load, resource accessability weren't explored as well. 

Areas for future research should focus more on longitudinal and experimental studies testing and for confounding variable and overall clarifying the concept, its effect, inclusions and exclusion to have overall more replicable and accurate studies. Also, exploring the malleability of grit, its cross-cultural, gender, socioeconomic influences. As well as, looking at areas outside of academic achievement, such as sports and work.

Although the research on grit and academic achievement is conflicting and confusing, developing grit is a fantastic was to practice stamina and can be applicable to all facets of life{{rewrite}}.

{{robelbox|theme=11|title=Test yourself!|iconwidth=55px|icon=Search-icon-white-background.png}} 
&lt;quiz display=simple&gt;
{What is Grit?}
- Cognitive trait
+ non-cognitive trait
+ Perseverance of Effort and Consistency of Interest
- Adaptability and Commitment

{How does grit influence academic achievement?}
+ persist through setbacks and sustain effort toward goals
- try and persist until faced with a difficult challenge
- use brute force or rote learning to facilitate learning
+ self-regulated learning strategies or deliberate practice

{What psychological theories explain grit?}
+ Social-determination theory and Locke theory of goal setting
- Adaptability 2-factor theory and Interest &amp; persistence theory
+ Cloninger’s seven-factor psychobiological model of personality, Achievement goal theory
- Intellectual modification and Fixed Mindset theory

{What does the empirical evidence say about grit? Any critiques?}
- Grit is the sole predictor of academic success
+ The research on grit is confusing regarding it being the predictor of academic success

{Can grit be fostered?}
- Grit is viewed as a malleable which can be assisted using independant and fixed mindset
+ Grit is viewed as a malleable which can be assisted using coping strategies, growth mindset and support
&lt;/quiz&gt;{{RoundBoxBottom}}

{{robelbox|theme=11|title=Answers!|iconwidth=55px|icon=Search-icon-white-background.png}} 
What is Grit?
* non-cognitive trait
* Perseverance of Effort and Consistency of Interest

How does grit influence academic achievement?
* persist through setbacks and sustain effort toward goals
* self-regulated learning strategies or deliberate practice

What psychological theories explain grit?
* Social-determination theory and Locke theory of goal setting
* Cloninger’s seven-factor psychobiological model of personality, Achievement goal theory

What does the empirical evidence say about grit? Any critiques?
* The research on grit is confusing regarding it being the predictor of academic success

Can grit be fostered?
* Grit is viewed as a malleable which can be assisted using coping strategies, growth mindset and support{{RoundBoxBottom}}

== See also ==
* [[Motivation and emotion/Book/2021/Academic buoyancy|Academic buoyancy]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Academic resilience|Academic resilience]] (Wikiversity)
* [[Motivation and emotion/Book/2020/Conscientiousness and motivation|Conscientiousness and motivation]] (Wikiversity)
* [[Motivation and emotion/Book/2020/Deliberate practice and mastery|Deliberate practice and mastery]] (Wikiversity)
* [[Motivation and emotion/Book/2019/Expectancy-value theory of achievement motivation|Expectancy-value theory of achievement motivation]] (Wikiversity)
* [[Motivation and emotion/Book/2020/Feedback and motivation in sport|Feedback and motivation in sport]] (Wikiversity)
* [[Motivation and emotion/Book/2024/Grit and conscientiousness|Grit and conscientiousness]]  (Wikiversity)
* [[Motivation and emotion/Book/2018/Growth mindset development|Growth mindset development]] (Wikiversity)
* [[Motivation and emotion/Book/2017/Hardiness|Hardiness]] (Wikiversity)
* [[Motivation and emotion/Book/2016/Long-term goal achievement|Long-term goal achievement]]  (Wikiversity)
* [[Motivation and emotion/Book/2021/Mental toughness|Mental toughness]] (Wikiversity)
* [[Motivation and emotion/Book/Chapters by year|Motivation and emotion book chapters]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Perseverance|Perseverance]] (Wikiversity)
* [[Motivation and emotion/Book/2022/Self-efficacy and academic achievement|Self-efficacy and academic achievement]] (Wikiversity)
* [[Motivation and emotion/Book/2015/Willpower|Willpower]] (Wikiversity)

== References ==
{{Hanging indent|1=

Abubakar, U., Azli, N. A. S. M., Hashim, I. A., Kamarudin, N. F. A., Latif, N. A. I. A., Badaruddin, A. R. M., Razak, M. Z., &amp; Zaidan, N. A. (2021). Association between grit and academic achievement among undergraduate pharmacy students in Malaysia. Currents in pharmacy teaching and learning, 13(5), 550-555. https://doi.org/10.1016/j.cptl.2021.01.013
Barbouta, A., Barbouta, C., &amp; Kotrotsiou, S. (2020). Growth Mindset and Grit: How Do University Students' Mindsets and Grit Affect their Academic Achievement? International journal of caring sciences, 13(1), 654-664. https://www.internationaljournalofcaringsciences.org/docs/72.%20kotrotsiou%206-2-2020.pdf

Clark, K. N., &amp; Malecki, C. K. (2019). Academic Grit Scale: Psychometric properties and associations with achievement and life satisfaction. Journal of School Psychology, 72, 49-66. https://doi.org/10.1016/j.jsp.2018.12.001

Clark, K. N., Dorio, N. B., Eldridge, M. A., Malecki, C. K., &amp; Demaray, M. K. (2020). Adolescent academic achievement: A model of social support and grit. Psychology in the schools, 57(2), 204-221. https://doi.org/10.1002/pits.22318

Deci, E., &amp; Ryan, R. (2000). ''Self-Determination Theory - an overview {{!}} ScienceDirect Topics.'' Sciencedirect.com. https://www.sciencedirect.com/topics/social-sciences/self-determination-theory

Hamdy Helal, M., &amp; Abohashem Hassan, E. (2025). Research motivation as a mediating variable between system intelligence, academic grit, and academic achievement among postgraduate students, faculty of education, Zagazig University. BMC Psychology, 13(1), 70-78. https://doi.org/10.1186/s40359-025-02374-z

Harpaz, G., Vaizman, T., &amp; Yaffe, Y. (2024). University students' academic grit and academic achievements predicted by subjective well‐being, coping resources, and self‐cultivation characteristics. Higher education quarterly, 78(1), 192-211. https://doi.org/10.1111/hequ.12455

Harwood, C., &amp; Thrower, S. (2020). Achievement Goal Theory - an overview {{!}} ScienceDirect Topics. Www.sciencedirect.com. https://www.sciencedirect.com/topics/psychology/achievement-goal-theory

He, X., Wang, H., Chang, F., Dill, S.-E., Liu, H., Tang, B., &amp; Shi, Y. (2021). IQ, grit, and academic achievement: Evidence from rural China. International journal of educational development, 80, 102306. https://doi.org/10.1016/j.ijedudev.2020.102306

Jiang, W., Xiao, Z., Liu, Y., Guo, K., Jiang, J., &amp; Du, X. (2019). Reciprocal relations between grit and academic achievement: A longitudinal study. Learning and Individual Differences, 71, 13-22. https://doi.org/10.1016/j.lindif.2019.02.004

Lam, K. K. L., &amp; Zhou, M. (2019). Examining the relationship between grit and academic achievement within K‐12 and higher education: A systematic review. Psychology in the schools, 56(10), 1654-1686. https://doi.org/10.1002/pits.22302

Lam, K. K. L., &amp; Zhou, M. (2022). Grit and Academic Achievement: A Comparative Cross-Cultural Meta-Analysis. Journal of educational psychology, 114(3), 597-621. https://doi.org/10.1037/edu0000699

Lee, S., &amp; Sohn, Y. W. (2017). Effects of grit on academic achievement and career-related attitudes of college students in Korea. Social behavior and personality, 45(10), 1629-1642. https://doi.org/10.2224/sbp.6400

Lee, S., Kim, Y., &amp; Shin, J. (2025). Exploring the Interplay of Social Support, Grit, and Achievement in Korean Junior High School Students. Psychology in the schools, 62(7), 2300-2310. https://doi.org/10.1002/pits.23467

Martoyo, I., &amp; Lindawati, L. (2023). Grit, Student Academic Achievement and Factors Affecting It. Jurnal psikologi teori dan terapan (Online), 14(3), 262-269. https://doi.org/10.26740/jptt.v14n03.p262-269

‌O’Connor, M. C., &amp; Paunonen, S. V. (2007). Big Five personality predictors of post-secondary academic performance. Personality and Individual Differences, 43(5), 971–990. https://doi.org/10.1016/j.paid.2007.03.017

Rimfeld, K., Kovas, Y., Dale, P. S., &amp; Plomin, R. (2016). True Grit and Genetics: Predicting Academic Achievement From Personality. Journal of personality and social psychology, 111(5), 780-789. https://doi.org/10.1037/pspp0000089

Segal, H., &amp; Kalfon-Hakhmigari, M. (2025). Grit as a moderator of the association between cognitive abilities and STEM achievements in high school. International journal of STEM education, 12(1), 25-14. https://doi.org/10.1186/s40594-025-00536-4

Singh, S., &amp; Chukkali, S. (2021). Development and validation of multi-dimensional scale of grit. *Cogent psychology, 8*(1). https://doi.org/10.1080/23311908.2021.1923166

Tannoubi, A., Quansah, F., Magouri, I., Chalghaf, N., Bonsaksen, T., Srem-Sai, M., Hagan, J. E., Handrianto, C., Azaiez, F., &amp; Bragazzi, N. L. (2023). Modelling the associations between academic engagement, study process and grit on academic achievement of physical education and sport university students. BMC Psychology, 11(1), 1-9. https://doi.org/10.1186/s40359-023-01454-2

Tugabirwe, I., &amp; Rukundo, A. (2024). Grit Predicts Academic Achievement among Undergraduate Science Teachers at a University of Science and Technology. Qeios, 6(1). https://doi.org/10.32388/MMPITX.2

Wolters, C. A., &amp; Hussain, M. (2015). Investigating grit and its relations with college students’ self-regulated learning and academic achievement. *Metacognition and learning, 10*(3), 293-311. https://doi.org/10.1007/s11409-014-9128-9
}}

== External links ==
* [https://penntoday.upenn.edu/news/lesson-grit-angela-duckworth A lesson in grit from Angela Duckworth]  (Penn Today)
* [https://theconversation.com/do-we-actually-grow-from-adversity-122252 Do we actually grow from adversity?]  (The Conversation)
* [https://theconversation.com/grit-matters-when-a-child-is-learning-to-read-even-in-poor-south-african-schools-157982 Grit matters when a child is learning to read, even in poor South African schools] (The Conversation)
* [https://theconversation.com/grit-or-quit-how-to-help-your-child-develop-resilience-195195 Grit or quit? How to help your child develop resilience] (The Conversation)
* [https://www.bbc.com/worklife/article/20210601-grit-the-dark-side-of-deciding-to-tough-it-out Grit: The dark side of deciding to 'tough it out']  (ABC)
* [https://www.youtube.com/watch?v=W-ONEAcBeTk Grit: The Power of Passion and Perseverance | Angela Duckworth | Talks at Google] (YouTube)
* [https://www.youtube.com/watch?v=H14bBuluwB8 Grit: The Power of Passion and Perseverance | Angela Lee Duckworth | TED] (YouTube)
* [https://www.forbes.com/sites/angelicagutierrez/2025/07/10/grit-why-it-matters-and-how-to-develop-it/ Grit: Why It Matters And How To Develop It]  (Forbes)
* [https://theconversation.com/grit-and-relentless-perseverance-can-take-a-toll-on-brain-health-particularly-for-people-facing-social-stresses-like-racism-251585 ‘Grit’ and relentless perseverance can take a toll on brain health − particularly for people facing social stresses like racism] (The Conversation)
* [https://www.abc.net.au/news/2018-07-31/growth-mindset-grit-and-resilience-key-to-success/10055608 Growth mindset, grit and resilience are central to getting what you want, psychologists say] (ABC)
* [https://www.forbes.com/sites/joanmichelson2/2018/08/30/high-achievers-have-more-grit-than-talent-8-ways-you-can-too/ High Achievers Have More Grit Than Talent] (Forbes)
* [https://www.theguardian.com/society/2025/may/21/its-not-grit-that-children-lack-but-proper-support It’s not ‘grit’ that children lack, but proper support]  (The Guardian)
* [https://theconversation.com/psychological-grit-is-over-rated-as-the-key-to-retention-in-distance-education-a-south-african-study-debunks-the-myth-199022 Psychological grit is over-rated as the key to retention in distance education: a South African study debunks the myth]  (The Conversation)
* [https://www.youtube.com/watch?v=cgLG9bFYmZc Should Schools Be Teaching Kids ‘Grit’ to Fix Mental Health? | This Morning's View]  (YouTube)
* [https://theconversation.com/true-grit-we-measured-it-and-found-it-protected-doctors-from-career-burnout-170628 True grit – we measured it and found it protected doctors from career burnout] (The Conversation)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Achievement]]
[[Category:Motivation and emotion/Book/Grit] None</text>
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  <page>
    <title>Introduction to thermodynamics</title>
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      <text bytes="4556" sha1="2ogovxvrwaaxlb4klcbl58bn11ko385" xml:space="preserve">This is the introduction course for thermodynamics, which is located in the [[Thermodynamics|topic of thermodynamics]].

Thermodynamics is the study of energy, energy transformation and their interactions with matter. The application of thermodynamics enables analysis of a wide range of technologies such as steam engines, refrigerators and turbines. &quot;Thermodynamics&quot; comes from the Greek words &quot;therme&quot; which means heat and &quot;dynamikos&quot; which means force, or power. So, &quot;Thermodynamics&quot; is the study of forces due to heat or heat due to forces. It deals with the conversion of energies through various forms and to various systems, as well as energies' relationships with the properties of a system, process and cycles.

== Chapters ==

# Basic concepts and definitions
## [[Introduction to thermodynamics/Introduction|Introduction]]
## [[Introduction to thermodynamics/What is thermodynamics?|What is thermodynamics?]]
## [[Introduction to thermodynamics/System and surroundings|System and surroundings]]
## [[Introduction to thermodynamics/Extensive and intensive properties|Extensive and intensive properties]]
## [[Introduction to thermodynamics/State postulate|State postulate]]
## [[Introduction to thermodynamics/State, process, and cycle|State, process, and cycle]]
## [[Introduction to thermodynamics/Temperature and heat|Temperature and heat]]
## [[Introduction to thermodynamics/First chapter review|First chapter review]]
# Thermodynamic properties of a pure substance
## [[Introduction to thermodynamics/Chapter introduction and learning objectives|Chapter introduction and learning objectives]]
## [[Introduction to thermodynamics/Pure substances|Pure substances]]
## Thermodynamic properties
## [[Introduction to thermodynamics/Phase diagrams|Phase diagrams]]
## Thermodynamic tables
## Chapter review
## Key Equations
# Real and ideal gases
## [[Introduction to thermodynamics/Introduction to real and ideal gases|Introduction to real and ideal gases]]
## Ideal gas and ideal gas equation of state
## Real gas and compressibility factor
## Chapter review
## Key equations
#First law of thermodynamics for closed systems
##[[Chapter introduction and learning objectives
##Internal energy in a system
##Heat transfer across a boundary
##Work
## [[Introduction to thermodynamics/The first law of thermodynamics for closed systems|The first law of thermodynamics for closed systems]]
## [[Introduction to thermodynamics/Chapter review|Chapter review]]
##Key equations
#The first law of thermodynamics for a control volume
##Chapter introduction and learning objectivews
##Enthalpy
##Mass and energy conservation equations in a control volume
##Applications of a the mass and energy conservation equations in steady flow devices
##Chapter review
##Key equations
#Entropy and the second law of thermodynamics
##[[Introduction to thermodynamics/Chapter introduction and learning objectives|Chapter introduction and learning objectives]]
##Heat engine
##[[Introduction to thermodynamics/Refrigerator and heat pump|Refrigerator and heat pump]]
##[[Introduction to thermodynamics/The second law of thermodynamics: Kelvin-Planck and Clausius statements|The second law of thermodynamics: Kelvin-Planck and Clausius statements]]
##[[Introduction to thermodynamics/Carnot cycles|Carnot cycles]]
##[[Introduction to thermodynamics/Entropy and entropy generation|Entropy and entropy generation]]
##The second law of thermodynamics for closed systems
##Specific entropy of a state
##Applications of the second law of thermodynamics in closed systems
##The second law of thermodynamics for open systems
##[[Introduction to thermodynamics/Applications of the second law of thermodynamics in open systems|Applications of the second law of thermodynamics in open systems]]
##Chapter reivew
##Key equations
#Absolute zero and the third law of thermodynamics
##Chapter introduction and learning objectives
##[[Introduction to thermodynamics/The third law of thermodynamics: Nernst principle|The third law of thermodynamics: Nernst principle]]
##[[Introduction to thermodynamics/Applications of the third law of dynamics|Applications of the third law of dynamics]]

== Attribution ==
Parts of this course incorporate material from the [https://pressbooks.bccampus.ca/thermo1 Introduction to Engineering Thermodynamics] course published by Pressbooks.&lt;ref&gt;  Yu Yan, C. (2022). Introduction to Engineering Thermodynamics. BCcampus. https://pressbooks.bccampus.ca/thermo1/&lt;/ref&gt;

== References ==
[[Category:Courses]]
[[Category:Science]]
[[Category:Thermodynamics]]</text>
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      <text bytes="4552" sha1="gzxkywnq9rjkk721votcn5xwqwqy1oy" xml:space="preserve">This is the introduction course for thermodynamics, which is located in the [[Thermodynamics|topic of thermodynamics]].

Thermodynamics is the study of energy, energy transformation and their interactions with matter. The application of thermodynamics enables analysis of a wide range of technologies such as steam engines, refrigerators and turbines. &quot;Thermodynamics&quot; comes from the Greek words &quot;therme&quot; which means heat and &quot;dynamikos&quot; which means force, or power. So, &quot;Thermodynamics&quot; is the study of forces due to heat or heat due to forces. It deals with the conversion of energies through various forms and to various systems, as well as energies' relationships with the properties of a system, process and cycles.

== Chapters ==

# Basic concepts and definitions
## [[Introduction to thermodynamics/Introduction|Introduction]]
## [[Introduction to thermodynamics/What is thermodynamics?|What is thermodynamics?]]
## [[Introduction to thermodynamics/System and surroundings|System and surroundings]]
## [[Introduction to thermodynamics/Extensive and intensive properties|Extensive and intensive properties]]
## [[Introduction to thermodynamics/State postulate|State postulate]]
## [[Introduction to thermodynamics/State, process, and cycle|State, process, and cycle]]
## [[Introduction to thermodynamics/Temperature and heat|Temperature and heat]]
## [[Introduction to thermodynamics/Chapter one review|Chapter one review]]
# Thermodynamic properties of a pure substance
## [[Introduction to thermodynamics/Chapter introduction and learning objectives|Chapter introduction and learning objectives]]
## [[Introduction to thermodynamics/Pure substances|Pure substances]]
## Thermodynamic properties
## [[Introduction to thermodynamics/Phase diagrams|Phase diagrams]]
## Thermodynamic tables
## Chapter review
## Key Equations
# Real and ideal gases
## [[Introduction to thermodynamics/Introduction to real and ideal gases|Introduction to real and ideal gases]]
## Ideal gas and ideal gas equation of state
## Real gas and compressibility factor
## Chapter review
## Key equations
#First law of thermodynamics for closed systems
##[[Chapter introduction and learning objectives
##Internal energy in a system
##Heat transfer across a boundary
##Work
## [[Introduction to thermodynamics/The first law of thermodynamics for closed systems|The first law of thermodynamics for closed systems]]
## [[Introduction to thermodynamics/Chapter review|Chapter review]]
##Key equations
#The first law of thermodynamics for a control volume
##Chapter introduction and learning objectivews
##Enthalpy
##Mass and energy conservation equations in a control volume
##Applications of a the mass and energy conservation equations in steady flow devices
##Chapter review
##Key equations
#Entropy and the second law of thermodynamics
##[[Introduction to thermodynamics/Chapter introduction and learning objectives|Chapter introduction and learning objectives]]
##Heat engine
##[[Introduction to thermodynamics/Refrigerator and heat pump|Refrigerator and heat pump]]
##[[Introduction to thermodynamics/The second law of thermodynamics: Kelvin-Planck and Clausius statements|The second law of thermodynamics: Kelvin-Planck and Clausius statements]]
##[[Introduction to thermodynamics/Carnot cycles|Carnot cycles]]
##[[Introduction to thermodynamics/Entropy and entropy generation|Entropy and entropy generation]]
##The second law of thermodynamics for closed systems
##Specific entropy of a state
##Applications of the second law of thermodynamics in closed systems
##The second law of thermodynamics for open systems
##[[Introduction to thermodynamics/Applications of the second law of thermodynamics in open systems|Applications of the second law of thermodynamics in open systems]]
##Chapter reivew
##Key equations
#Absolute zero and the third law of thermodynamics
##Chapter introduction and learning objectives
##[[Introduction to thermodynamics/The third law of thermodynamics: Nernst principle|The third law of thermodynamics: Nernst principle]]
##[[Introduction to thermodynamics/Applications of the third law of dynamics|Applications of the third law of dynamics]]

== Attribution ==
Parts of this course incorporate material from the [https://pressbooks.bccampus.ca/thermo1 Introduction to Engineering Thermodynamics] course published by Pressbooks.&lt;ref&gt;  Yu Yan, C. (2022). Introduction to Engineering Thermodynamics. BCcampus. https://pressbooks.bccampus.ca/thermo1/&lt;/ref&gt;

== References ==
[[Category:Courses]]
[[Category:Science]]
[[Category:Thermodynamics]]</text>
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      <text bytes="4588" sha1="bpd9yd2l39pe6daple1xtwy1leprjyw" xml:space="preserve">This is the introduction course for thermodynamics, which is located in the [[Thermodynamics|topic of thermodynamics]].

Thermodynamics is the study of energy, energy transformation and their interactions with matter. The application of thermodynamics enables analysis of a wide range of technologies such as steam engines, refrigerators and turbines. &quot;Thermodynamics&quot; comes from the Greek words &quot;therme&quot; which means heat and &quot;dynamikos&quot; which means force, or power. So, &quot;Thermodynamics&quot; is the study of forces due to heat or heat due to forces. It deals with the conversion of energies through various forms and to various systems, as well as energies' relationships with the properties of a system, process and cycles.

== Chapters ==

# Basic concepts and definitions
## [[Introduction to thermodynamics/Introduction|Introduction]]
## [[Introduction to thermodynamics/What is thermodynamics?|What is thermodynamics?]]
## [[Introduction to thermodynamics/System and surroundings|System and surroundings]]
## [[Introduction to thermodynamics/Extensive and intensive properties|Extensive and intensive properties]]
## [[Introduction to thermodynamics/State postulate|State postulate]]
## [[Introduction to thermodynamics/State, process, and cycle|State, process, and cycle]]
## [[Introduction to thermodynamics/Temperature and heat|Temperature and heat]]
## [[Introduction to thermodynamics/Chapter one review|Chapter one review]]
# Thermodynamic properties of a pure substance
## [[Introduction to thermodynamics/Chapter introduction and learning objectives|Chapter introduction and learning objectives]]
## [[Introduction to thermodynamics/Pure substances|Pure substances]]
## Thermodynamic properties
## [[Introduction to thermodynamics/Phase diagrams|Phase diagrams]]
## Thermodynamic tables
## Chapter review
## Key Equations
# Real and ideal gases
## [[Introduction to thermodynamics/Introduction to real and ideal gases|Introduction to real and ideal gases]]
## Ideal gas and ideal gas equation of state
## Real gas and compressibility factor
## Chapter review
## Key equations
# First law of thermodynamics for closed systems
## Chapter introduction and learning objectives
## Internal energy in a system
## Heat transfer across a boundary
## Work
## [[Introduction to thermodynamics/The first law of thermodynamics for closed systems|The first law of thermodynamics for closed systems]]
## [[Introduction to thermodynamics/Chapter four review|Chapter four review]]
##Key equations
# The first law of thermodynamics for a control volume
## Chapter introduction and learning objectives
## Enthalpy
## Mass and energy conservation equations in a control volume
## Applications of a the mass and energy conservation equations in steady flow devices
## Chapter review
## Key equations
# Entropy and the second law of thermodynamics
## [[Introduction to thermodynamics/Chapter introduction and learning objectives|Chapter introduction and learning objectives]]
## Heat engine
## [[Introduction to thermodynamics/Refrigerator and heat pump|Refrigerator and heat pump]]
## [[Introduction to thermodynamics/The second law of thermodynamics: Kelvin-Planck and Clausius statements|The second law of thermodynamics: Kelvin-Planck and Clausius statements]]
## [[Introduction to thermodynamics/Carnot cycles|Carnot cycles]]
## [[Introduction to thermodynamics/Entropy and entropy generation|Entropy and entropy generation]]
## The second law of thermodynamics for closed systems
## Specific entropy of a state
## Applications of the second law of thermodynamics in closed systems
## The second law of thermodynamics for open systems
## [[Introduction to thermodynamics/Applications of the second law of thermodynamics in open systems|Applications of the second law of thermodynamics in open systems]]
## Chapter reivew
## Key equations
#Absolute zero and the third law of thermodynamics
## Chapter introduction and learning objectives
## [[Introduction to thermodynamics/The third law of thermodynamics: Nernst principle|The third law of thermodynamics: Nernst principle]]
## [[Introduction to thermodynamics/Applications of the third law of dynamics|Applications of the third law of dynamics]]

== Attribution ==
Parts of this course incorporate material from the [https://pressbooks.bccampus.ca/thermo1 Introduction to Engineering Thermodynamics] course published by Pressbooks.&lt;ref&gt;  Yu Yan, C. (2022). Introduction to Engineering Thermodynamics. BCcampus. https://pressbooks.bccampus.ca/thermo1/&lt;/ref&gt;

== References ==
[[Category:Courses]]
[[Category:Science]]
[[Category:Thermodynamics]]</text>
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    <title>OpenStax Astronomy 2e</title>
    <ns>0</ns>
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      <parentid>2806302</parentid>
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        <id>276019</id>
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      <text bytes="547" sha1="c54fi61kfqsw122j9k7nbwldq4rwq9e" xml:space="preserve">See also [[OpenStax Astronomy]] and [[OpenStax]]


'''&lt;big&gt;OpenStax Astronomy 2e&lt;/big&gt;'''

* [https://openstax.org/details/books/astronomy-2e OpenStax Astronomy 2e] (original content)
* [https://audileo.com/audiobooks/openstax/astronomy-2e/ OpenStax Astronomy 2e audiobook] (full audiobook)
* [https://www.youtube.com/watch?v=jYvKXcXIz-g OpenStax Astronomy 2e audiobook] (chapter 1)
* [https://www.youtube.com/watch?v=caCQf5o9mpI&amp;list=PLwoTa9G34hiOY7iOAGjURgOT86OPLlNRA Lecture videos] fo Astronomy 2e

[[Category:Astronomy]]
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'''&lt;big&gt;OpenStax A&lt;/big&gt;&lt;big&gt;natomy &amp; Physiology 2e&lt;/big&gt;'''

''Anatomy and Physiology 2e'' is developed to meet the scope and sequence for a two-semester human anatomy and physiology course for life science and allied health majors. The book is organized by body systems. The revision focuses on inclusive and equitable instruction and includes new student support. Illustrations have been extensively revised to be clearer and more inclusive. The web-based version of ''Anatomy and Physiology 2e'' also features links to surgical videos, histology, and interactive diagrams. Please learn more about the changes by previewing the preface.

* [https://openstax.org/details/books/anatomy-and-physiology-2e OpenStax Anatomy &amp; Physiology 2e] (original content)
* [https://audileo.com/audiobooks/openstax/anatomy-physiology-2e/ OpenStax Anatomy &amp; Physiology 2e audiobook] (official audiobook)
* [https://www.youtube.com/watch?v=rjQ5WS8XeBU OpenStax Anatomy &amp; Physiology 2e audio textbook] (chapter 1 on YouTube)
* [https://www.youtube.com/watch?v=BhmhqyNzLls Lecture videos] for A&amp;P 2e

[[Category:Anatomy]]
[[Category:Physiology]]
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    <title>OpenStax Psychology 2e</title>
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'''&lt;big&gt;OpenStax Psychology 2e&lt;/big&gt;'''

* [https://openstax.org/details/books/psychology-2e OpenStax Psychology 2e] (original content)
* [https://audileo.com/audiobooks/openstax/psychology-2e/ OpenStax Psychology 2e audiobook] (full audiobook)
* Lecture videos of Psychology 2e on [https://www.youtube.com/watch?v=vzeI4foXX2M&amp;list=PLK3eIt61XPo36cXXo_JjSFLPztiDTdhk5&amp;index=1 YouTube]
* [https://www.youtube.com/watch?v=X2sg48CSF2g Psychology 2e audiobook] (chapter 1)

[[Category:Psychology]]
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      <text bytes="589" sha1="pcx6px60jnav59a5bvjjyneu781xh9w" xml:space="preserve">See also [[OpenStax American Government 4e|OpenStax American Government]] [[OpenStax American Government 4e|4e]] (2025 edition)

See also [[OpenStax]]

'''&lt;big&gt;OpenStax American Government 3e&lt;/big&gt;'''

* [https://openstax.org/details/books/american-government-3e OpenStax American Government 3e] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/american-government-3e/ OpenStax American Government 3e audiobook] Available as audio textbook.
* Lecture videos for American Government 3e

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        <id>276019</id>
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See also [[OpenStax]]


'''&lt;big&gt;OpenStax American Government&lt;/big&gt;''' '''&lt;big&gt;4e&lt;/big&gt;'''

This publiscation was released in July 2025 and includes updated information on the 2024 election results, recent court decisions and legal actions, and new opportunities for civic engagement.

* [https://openstax.org/details/books/american-government-4e OpenStax American Government 4e] (original content)
* [https://audileo.com/audiobooks/openstax/american-government-4e/ OpenStax American Government 4e audiobook] (full audiobook)
* [https://www.youtube.com/watch?v=4udvBo3C-qM American Government 4e audiobook] (chapter 1 on YouTube)

[[Category:United States Government]]
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    <title>OpenStax Biology 2e</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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== OpenStax Biology 2e ==

* [https://openstax.org/details/books/biology-2e OpenStax Biology 2e] (original content)
* [https://audileo.com/audiobooks/openstax/biology-2e/ OpenStax Biology 2e audiobook]
* Lecture videos for Biology 2e

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== OpenStax U.S. History ==
''U.S. History'' is designed to meet the scope and sequence requirements of most introductory courses. The text provides a balanced approach to U.S. history, considering the people, events, and ideas that have shaped the United States from both the top down (politics, economics, diplomacy) and bottom up (eyewitness accounts, lived experience). ''U.S. History'' covers key forces that form the American experience, with particular attention to issues of race, class, and gender.

* [https://openstax.org/details/books/us-history OpenStax U.S. History] (original content)
* [https://audileo.com/audiobooks/openstax/us-history/ OpenStax U.S. History audiobook]
* [https://www.youtube.com/watch?v=v7B8Gn17qk8&amp;list=PLPvvRbQA5ODMB4vDRdRF66I9m43TMVyx3&amp;index=1 Lecture videos] for U.S. History

[[Category:History of the United States]]
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== OpenStax Introduction to Political Science ==
Designed to meet the scope and sequence of your course, OpenStax ''Introduction to Political Science'' provides a strong foundation in global political systems, exploring ''how'' and ''why'' political realities unfold. Rich with examples of individual and national social action, this text emphasizes students’ role in the political sphere and equips them to be active and informed participants in civil society.

* [https://openstax.org/details/books/introduction-political-science OpenStax Introduction to Political Science] (original content)
* [https://audileo.com/audiobooks/openstax/introduction-to-political-science/ OpenStax Introduction to Political Science Audiobook]

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'''&lt;big&gt;OpenStax&lt;/big&gt;''' '''&lt;big&gt;OpenStax World History, Volume 1: to 1500&lt;/big&gt;'''

* [https://openstax.org/details/books/world-history-volume-1 OpenStax OpenStax World History, Volume 1: to 1500] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/world-history-volume-1-to-1500/ OpenStax OpenStax World History, Volume 1: to 1500 audiobook] Available as audio textbook.

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    <title>OpenStax Concepts of Biology</title>
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'''&lt;big&gt;OpenStax&lt;/big&gt;''' '''&lt;big&gt;Concepts of Biology&lt;/big&gt;'''

* [https://openstax.org/details/books/concepts-biology OpenStax Concepts of Biology] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/concepts-of-biology/ OpenStax Concepts of Biology audiobook] Available as audio textbook.

[[Category:Biology]]
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    <title>OpenStax Lifespan Development</title>
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== '''&lt;big&gt;OpenStax&lt;/big&gt;''' '''&lt;big&gt;Lifespan Development&lt;/big&gt;''' ==

== Summary ==
''Lifespan Development'' aligns to the topics and objectives of most introductory developmental psychology courses taught across departments. Grounded in foundational theories and scientific research, the text teaches students about core aspects of human development—physical, cognitive, social, emotional—across the lifespan. A primary goal of the book is to incorporate content, scholarship, and activities that explore a variety of perspectives that encourage all students to feel seen and included.

''Lifespan Development'' strives to openly address complex topics with scholarly responsibility and an effort to increase equity and inclusion in the research presented, as well as to foster student engagement in the classroom through relevant examples and applications. Focused on driving meaningful and memorable learning experiences, the narrative places concepts in contexts that give students the means to understand human development and how that knowledge can be applied to and improve their own lives and the lives of others.

* [https://openstax.org/details/books/lifespan-development OpenStax Lifespan Development] (original content).  Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/lifespan-development/ OpenStax Lifespan Development audiobook]  Available as audio textbook.

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      <text bytes="7181" sha1="2puc1j7907008s0nf82dq7fjtbdymeu" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = The second law of thermodynamics: Kelvin-Planck and Clausius statements
 | previous= Refrigerator and heat pump
 | next    = Carnot cycles
}}

== Introduction ==
The 1st Law of Thermodynamics tells us that an increase in one form of energy, ''E,'' must be accompanied by a decrease in another form of energy, ''E''. Likewise the 2nd Law of Thermodynamics tells us which other kinds of processes in nature may or may not occur. For instance, with two objects in thermal contact, heat will spontaneously flow from a warmer to cooler object, but this effect does not spontaneously occur in the opposite direction. As with other physical laws, the two possible statements of the second law are based on experimental observations.
* [[w:_Irreversible_process|Irreversible process]] = processes that occur naturally in only one direction.
* [[w:_Reversible_process_(thermodynamics)|Reversible process]] = process that can occur in the reverse direction in infinitesimal steps

== Why a second law is needed ==
Suppose you have two separate containers each containing 1 kg of water. One container is at 20°C and the other is at 10°C. What temperature will the water be if you mix it? Can you prove it with the first law of thermodynamics?
[[File:Impossible_process_thermodynamics.drawio.svg|right|frameless|250x250px|Impossible thermodynamic process]]
As you can see, with just the use of the first law of thermodynamics, both of these scenarios are possible. However, intuitively, for this simple example, we know that this will not happen. The process will only end up at 15°C. 
{{Blockquote|text=It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.|author=Kelvin (based on Clausius statement).}}
In addition, if there is only reservoir a fluid, and there is no other reservoir which is at a separate temperature, no work or exchange of energy will naturally occur.
[[File:Impossible_cycle_thermodynamics_one_source.drawio.svg|left|frameless|Impossible thermodynamic process of only one source being used to produce work.]]
{{Blockquote|text=A cycle in contact with only one source cannot produce any work.|author=Kelvin-Planck principle.}}


It can be seen that at least two reservoirs, a relatively hotter one, and a relatively colder reservoir are need to drive a process to produce any work.
== Heat engines ==

[[Image:Enginos.jpg]]


Device that converts thermal energy, &lt;math display=&quot;inline&quot;&gt;E&lt;/math&gt; (i.e. &lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) into mechanical or electrical &lt;math display=&quot;inline&quot;&gt;E&lt;/math&gt;.

During the cycle heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is absorbed from a source at a high &lt;math&gt;T &lt;/math&gt; and work, &lt;math display=&quot;inline&quot;&gt;W&lt;/math&gt;, is done by the engine.

Heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is expelled to a source at a lower &lt;math&gt;T &lt;/math&gt;.

Since it is a cyclic process, &lt;math&gt;\Delta U = 0 &lt;/math&gt;. Thus, &lt;math&gt;W = Q_{total} \longrightarrow W = Q_h - Q_c &lt;/math&gt; 

&lt;math display=&quot;inline&quot;&gt;\eta = \text {efficiency} = \text {ratio of work done to heat absorbed}&lt;/math&gt;

&lt;math display=&quot;inline&quot;&gt;\eta = W = Q_h = \frac {(Q_h - Q_c)} { Q_h } = 1 - \frac {Q_c}{Q_h}&lt;/math&gt;

&lt;math display=&quot;inline&quot;&gt;\eta = 100%&lt;/math&gt; only if &lt;math display=&quot;inline&quot;&gt;Q_c = 0&lt;/math&gt;, i.e. a perfect heat engine would convert all of the absorbed heat &lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt; into mechanical work. From the 2nd law of thermodynamics, this is impossible: only a fraction of heat is converted to mechanical E.

E.g. η of automobile engines = 20%.

{{Blockquote|text=It is impossible by a cycle to take heat from a hot reservoir and convert it into work without, at the same time, transferring an amount of heat from hot to cold reservoir.|author=Kelvin–Planck form of the 2nd law of thermodynamics.}}

== The thermodynamic temperature scale (absolute T) ==

Since the efficiency of a Carnot engine is:

&lt;math&gt;\eta = 1 - \frac{Q_c}{Q_h} = 1 - \frac{T_c}{T_h}&lt;/math&gt;

The zero point of the thermodynamic scale must be fixed as the '''&lt;math&gt;T&lt;/math&gt;''' of the cold reservoir '''&lt;math&gt;T_c&lt;/math&gt;''' at which η = 1 and hence '''&lt;math&gt;Q_c = 0&lt;/math&gt;''' and '''&lt;math&gt;W = Q_h&lt;/math&gt;'''. T must be the absolute scale because otherwise it may have –ve values, and hence the engine will perform work more than the heat given by the source. i.e. we would create E. from nothing, which is in contradiction to 1st law of thermodynamics. Thus, T = 0 is the lowest T in all scales i.e. the absolute zero.

== Quasi–static reversible process for an ideal gas ==

For an ideal gas undergoing a quasi–static reversible process from &lt;math&gt;T_i \and V_i&lt;/math&gt; to &lt;math&gt;T_f \and V_f&lt;/math&gt;, 

&lt;math&gt;dQ_r = dU + dW&lt;/math&gt;; &lt;math&gt;dQ = PdV&lt;/math&gt;.    Since it’s an ideal gas,

&lt;math&gt;dU = n C_v dT&lt;/math&gt;; &lt;math&gt;P = \frac {nRT}{V}&lt;/math&gt;. Thus,

&lt;math&gt;dQ_r = n C_v dT + \frac {nRTdV}{V}&lt;/math&gt;. Dividing by &lt;math&gt;T&lt;/math&gt;,

&lt;math&gt;\frac {dQ_r}{T} = n C_v \frac {dT}{T} + nRT \frac {dV}{V}&lt;/math&gt;. Assuming &lt;math&gt;C_v&lt;/math&gt; is constant, 

by integration, &lt;math&gt;\Delta S = \int \frac {dQ_r}{T} = n C_v \ln (\frac {T_f}{T_i}) + n R \ln (\frac {V_f}{V_i})&lt;/math&gt;. Thus, &lt;math&gt;\Delta S&lt;/math&gt; is independent of the reversible path and depends only on initial and final states. For a cyclic process, &lt;math&gt;T_i = T_f&lt;/math&gt; and &lt;math&gt;V_i = V_f&lt;/math&gt;, so &lt;math&gt;\Delta S = 0&lt;/math&gt;.

== Heat conduction ==

* When heat transfers from a hot &lt;math&gt;T_h&lt;/math&gt; to cold &lt;math&gt;T_c&lt;/math&gt; reservoirs, the entropy of the cold reservoir increases by &lt;math&gt;\frac {Q}{T_c}&lt;/math&gt; and the entropy of the hot reservoir decreases by &lt;math&gt;\frac {Q}{T_h}&lt;/math&gt;.

* Since &lt;math&gt;T_c &lt; T_h&lt;/math&gt;, total change in entropy of the system (universe) &gt; 0

* &lt;math&gt;\Delta S_u = \frac {Q}{T_c} - \frac {Q}{T_h } &gt; 0&lt;/math&gt;
* Thus, if &lt;math&gt;\Delta S_u &lt; 0&lt;/math&gt;, the process cannot occur. For example heat transfer from cold to hot body will never occur without artificial assistance.

== Free expansion ==

[[Image:Expansionn.jpg]]

* An ideal gas in an insulated container occupies a volume &lt;math&gt;V_i&lt;/math&gt;. A membrane separating it from a vacuum is suddenly broken and the gas expands irreversibly to &lt;math&gt;V_f&lt;/math&gt;.
* Work done against the vacuum = 0. Since walls are perfectly insulated, &lt;math&gt;Q=0&lt;/math&gt;. Thus, &lt;math&gt;\Delta U =0&lt;/math&gt; and &lt;math&gt;U_i = U_f&lt;/math&gt;. Since the gas is ideal, &lt;math&gt;U&lt;/math&gt; depends only on &lt;math&gt;T&lt;/math&gt;. Thus, &lt;math&gt;T_i = T_f&lt;/math&gt;.
* Since &lt;math&gt;\Delta S = \int dS = \int \frac{dQ_r}{T}&lt;/math&gt; only applies to reversible processes, we cannot use it directly. Thus, we imagine a reversible process with the same initial and final states: an isothermal reversible expansion. Since T is constant, &lt;math&gt;\Delta S = \int \frac{dQ_r}{T} = \frac {1}{T} \int dQ_r&lt;/math&gt;.
* Since it is isothermal, &lt;math&gt;\int dQ_r = W = nRT \ln (\frac {V_f}{V_i})&lt;/math&gt;. Thus,

&lt;math&gt;\Delta S = n R \ln{(\frac {V_f}{V_i})}&lt;/math&gt; 

* Since &lt;math&gt;V_f &gt; V_i, \Delta S_u &gt; 0&lt;/math&gt;. This can also be obtained by equation *** by setting &lt;math&gt;T_f = T_i&lt;/math&gt;.

[[Category:Chemical engineering]]
[[Category:Thermodynamics]]</text>
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      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
      </contributor>
      <comment>equivalence of the two statements of the second law</comment>
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      <text bytes="7521" sha1="1rqyrj9dz0a2zrwlzy0czzndlt94e37" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = The second law of thermodynamics: Kelvin-Planck and Clausius statements
 | previous= Refrigerator and heat pump
 | next    = Carnot cycles
}}

== Introduction ==
The first law of thermodynamics tells us two facts verified by observations about our physical reality. The first is that an increase in one form of energy, ''E,'' must be accompanied by a decrease in another form of energy, ''E''. The second tells us which other kinds of processes in nature may or may not occur. For instance, with two objects in thermal contact, heat will spontaneously flow from a warmer to cooler object, but this effect does not spontaneously occur in the opposite direction. As with other physical laws, the two possible statements of the second law are based on experimental observations.
* [[w:_Irreversible_process|Irreversible process]] = processes that occur naturally in only one direction.
* [[w:_Reversible_process_(thermodynamics)|Reversible process]] = process that can occur in the reverse direction in infinitesimal steps

== Why a second law is needed ==
Suppose you have two separate containers each containing 1 kg of water. One container is at 20°C and the other is at 10°C. What temperature will the water be if you mix it? Can you prove it with the first law of thermodynamics?
[[File:Impossible_process_thermodynamics.drawio.svg|right|frameless|250x250px|Impossible thermodynamic process]]
As you can see, with just the use of the first law of thermodynamics, both of these scenarios are possible. However, intuitively, for this simple example, we know that this will not happen. The process will only end up at 15°C. 
{{Blockquote|text=It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.|author=Kelvin (based on Clausius statement).}}
In addition, if there is only reservoir a fluid, and there is no other reservoir which is at a separate temperature, no work or exchange of energy will naturally occur.
[[File:Impossible_cycle_thermodynamics_one_source.drawio.svg|left|frameless|Impossible thermodynamic process of only one source being used to produce work.]]
{{Blockquote|text=A cycle in contact with only one source cannot produce any work.|author=Kelvin-Planck principle.}}


It can be seen that at least two reservoirs, a relatively hotter one, and a relatively colder reservoir are need to drive a process to produce any work.
== Heat engines ==

[[Image:Enginos.jpg]]


Device that converts thermal energy, &lt;math display=&quot;inline&quot;&gt;E&lt;/math&gt; (i.e. &lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) into mechanical or electrical &lt;math display=&quot;inline&quot;&gt;E&lt;/math&gt;.

During the cycle heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is absorbed from a source at a high &lt;math&gt;T &lt;/math&gt; and work, &lt;math display=&quot;inline&quot;&gt;W&lt;/math&gt;, is done by the engine.

Heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is expelled to a source at a lower &lt;math&gt;T &lt;/math&gt;.

Since it is a cyclic process, &lt;math&gt;\Delta U = 0 &lt;/math&gt;. Thus, &lt;math&gt;W = Q_{total} \longrightarrow W = Q_h - Q_c &lt;/math&gt; 

&lt;math display=&quot;inline&quot;&gt;\eta = \text {efficiency} = \text {ratio of work done to heat absorbed}&lt;/math&gt;

&lt;math display=&quot;inline&quot;&gt;\eta = W = Q_h = \frac {(Q_h - Q_c)} { Q_h } = 1 - \frac {Q_c}{Q_h}&lt;/math&gt;

&lt;math display=&quot;inline&quot;&gt;\eta = 100%&lt;/math&gt; only if &lt;math display=&quot;inline&quot;&gt;Q_c = 0&lt;/math&gt;, i.e. a perfect heat engine would convert all of the absorbed heat &lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt; into mechanical work. From the 2nd law of thermodynamics, this is impossible: only a fraction of heat is converted to mechanical E.

E.g. η of automobile engines = 20%.

{{Blockquote|text=It is impossible by a cycle to take heat from a hot reservoir and convert it into work without, at the same time, transferring an amount of heat from hot to cold reservoir.|author=Kelvin–Planck form of the 2nd law of thermodynamics.}}

== Either statement independently supports the second law ==
Both the Kelvin-Planck and the Clausius statement are sufficiently independent to support the second law. This means that any cycle that violates the Kelvin-Planck statement, also violates the Clausius statement and vice-versa.

== The thermodynamic temperature scale (absolute T) ==

Since the efficiency of a Carnot engine is:

&lt;math&gt;\eta = 1 - \frac{Q_c}{Q_h} = 1 - \frac{T_c}{T_h}&lt;/math&gt;

The zero point of the thermodynamic scale must be fixed as the '''&lt;math&gt;T&lt;/math&gt;''' of the cold reservoir '''&lt;math&gt;T_c&lt;/math&gt;''' at which η = 1 and hence '''&lt;math&gt;Q_c = 0&lt;/math&gt;''' and '''&lt;math&gt;W = Q_h&lt;/math&gt;'''. T must be the absolute scale because otherwise it may have –ve values, and hence the engine will perform work more than the heat given by the source. i.e. we would create E. from nothing, which is in contradiction to 1st law of thermodynamics. Thus, T = 0 is the lowest T in all scales i.e. the absolute zero.

== Quasi–static reversible process for an ideal gas ==

For an ideal gas undergoing a quasi–static reversible process from &lt;math&gt;T_i \and V_i&lt;/math&gt; to &lt;math&gt;T_f \and V_f&lt;/math&gt;, 

&lt;math&gt;dQ_r = dU + dW&lt;/math&gt;; &lt;math&gt;dQ = PdV&lt;/math&gt;.    Since it’s an ideal gas,

&lt;math&gt;dU = n C_v dT&lt;/math&gt;; &lt;math&gt;P = \frac {nRT}{V}&lt;/math&gt;. Thus,

&lt;math&gt;dQ_r = n C_v dT + \frac {nRTdV}{V}&lt;/math&gt;. Dividing by &lt;math&gt;T&lt;/math&gt;,

&lt;math&gt;\frac {dQ_r}{T} = n C_v \frac {dT}{T} + nRT \frac {dV}{V}&lt;/math&gt;. Assuming &lt;math&gt;C_v&lt;/math&gt; is constant, 

by integration, &lt;math&gt;\Delta S = \int \frac {dQ_r}{T} = n C_v \ln (\frac {T_f}{T_i}) + n R \ln (\frac {V_f}{V_i})&lt;/math&gt;. Thus, &lt;math&gt;\Delta S&lt;/math&gt; is independent of the reversible path and depends only on initial and final states. For a cyclic process, &lt;math&gt;T_i = T_f&lt;/math&gt; and &lt;math&gt;V_i = V_f&lt;/math&gt;, so &lt;math&gt;\Delta S = 0&lt;/math&gt;.

== Heat conduction ==

* When heat transfers from a hot &lt;math&gt;T_h&lt;/math&gt; to cold &lt;math&gt;T_c&lt;/math&gt; reservoirs, the entropy of the cold reservoir increases by &lt;math&gt;\frac {Q}{T_c}&lt;/math&gt; and the entropy of the hot reservoir decreases by &lt;math&gt;\frac {Q}{T_h}&lt;/math&gt;.

* Since &lt;math&gt;T_c &lt; T_h&lt;/math&gt;, total change in entropy of the system (universe) &gt; 0

* &lt;math&gt;\Delta S_u = \frac {Q}{T_c} - \frac {Q}{T_h } &gt; 0&lt;/math&gt;
* Thus, if &lt;math&gt;\Delta S_u &lt; 0&lt;/math&gt;, the process cannot occur. For example heat transfer from cold to hot body will never occur without artificial assistance.

== Free expansion ==

[[Image:Expansionn.jpg]]

* An ideal gas in an insulated container occupies a volume &lt;math&gt;V_i&lt;/math&gt;. A membrane separating it from a vacuum is suddenly broken and the gas expands irreversibly to &lt;math&gt;V_f&lt;/math&gt;.
* Work done against the vacuum = 0. Since walls are perfectly insulated, &lt;math&gt;Q=0&lt;/math&gt;. Thus, &lt;math&gt;\Delta U =0&lt;/math&gt; and &lt;math&gt;U_i = U_f&lt;/math&gt;. Since the gas is ideal, &lt;math&gt;U&lt;/math&gt; depends only on &lt;math&gt;T&lt;/math&gt;. Thus, &lt;math&gt;T_i = T_f&lt;/math&gt;.
* Since &lt;math&gt;\Delta S = \int dS = \int \frac{dQ_r}{T}&lt;/math&gt; only applies to reversible processes, we cannot use it directly. Thus, we imagine a reversible process with the same initial and final states: an isothermal reversible expansion. Since T is constant, &lt;math&gt;\Delta S = \int \frac{dQ_r}{T} = \frac {1}{T} \int dQ_r&lt;/math&gt;.
* Since it is isothermal, &lt;math&gt;\int dQ_r = W = nRT \ln (\frac {V_f}{V_i})&lt;/math&gt;. Thus,

&lt;math&gt;\Delta S = n R \ln{(\frac {V_f}{V_i})}&lt;/math&gt; 

* Since &lt;math&gt;V_f &gt; V_i, \Delta S_u &gt; 0&lt;/math&gt;. This can also be obtained by equation *** by setting &lt;math&gt;T_f = T_i&lt;/math&gt;.

[[Category:Chemical engineering]]
[[Category:Thermodynamics]]</text>
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  <page>
    <title>Introduction to thermodynamics/Carnot cycles</title>
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    <revision>
      <id>2830574</id>
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      <timestamp>2026-09-02T20:06:46Z</timestamp>
      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
      </contributor>
      <comment>/* The Carnot Engine */</comment>
      <origin>2830574</origin>
      <model>wikitext</model>
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      <text bytes="3917" sha1="tgcf4ivvtdvle2aoqhfc6f6hhvwgmwm" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = Carnot cycles
 | previous= The second law of thermodynamics: Kelvin-Planck and Clausius statements
 | next    = Applications of the second law
}}

== The Carnot Engine ==
{{Defbox|name=Carnot's Theorem|value=Carnot’s theorem stated in two principles: The first is that no real (irreversible) engine operating between two heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same two reservoirs. The second is that any two reversible engines operating between the same two reservoirs, will have the same efficiency.|label=Theorem|italic=yes}}
* Carnot made the most efficient heat engine. Net work taken from the Carnot cycle is the largest possible for a given amount of heat supplied.
* Carnot’s theorem: No real (irreversible) engine operating between 2 heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same 2 reservoirs.
* Carnot used an ideal gas contained in a cylinder with a movable piston at one end. The cylinder walls and the piston are thermally non–conducting.
* The Carnot cycle consists of 2 adiabatic and 2 isothermal processes, all reversible:

'''&lt;math&gt;A \longrightarrow B&lt;/math&gt;'''

Isothermal expansion of a gas placed in thermal contact with a heat reservoir at temperature  '''&lt;math&gt;T_h&lt;/math&gt;'''.

During the process, the gas absorbs heat '''&lt;math&gt;Q_h&lt;/math&gt;''' from the base of the cylinder and does a work '''&lt;math&gt;WAB&lt;/math&gt;''' in raising the piston.

'''&lt;math&gt;B \longrightarrow C&lt;/math&gt;'''

Adiabatic expansion by replacing the base of the cylinder by a thermally non–conducting wall.

During the process, temperature, '''&lt;math&gt;T&lt;/math&gt;''',  falls from '''&lt;math&gt;T_h&lt;/math&gt;''' to '''&lt;math&gt;T_c&lt;/math&gt;''' and the gas does work '''&lt;math&gt;WBC&lt;/math&gt;''' in raising the piston.

'''&lt;math&gt;C \longrightarrow D&lt;/math&gt;'''

Isothermal compression by placing the gas in thermal contact with a heat reservoir at temp. Tc.

During the process, the gas expels heat '''&lt;math&gt;Q_c&lt;/math&gt;''' to the reservoir and the work '''&lt;math&gt;WCD&lt;/math&gt;''' is done on the gas by external agent.

'''&lt;math&gt;D \longrightarrow A&lt;/math&gt;'''

Adiabatic compression by replacing the base of the cylinder by a non–conducting wall.

During the process, temperature, '''&lt;math&gt;T&lt;/math&gt;''', increases from '''&lt;math&gt;T_c&lt;/math&gt;''' to '''&lt;math&gt;T_h&lt;/math&gt;''' and the work '''&lt;math&gt;WDA&lt;/math&gt;''' is done on the gas by external agent.

[[Image:Carnoti.jpg]]
[[File:Máquina_de_Carnot.gif|alt=Máquina de Carnot|left|thumb|500x500px]]
* Net work done in this reversible cyclic process = area enclosed by the path '''&lt;math&gt;ABCDA&lt;/math&gt;'''  = net heat transferred into the system, since '''&lt;math&gt;\Delta U = 0&lt;/math&gt;'''.
* Since the internal '''&lt;math&gt;E&lt;/math&gt;'''. of an ideal gas depends only on absolute '''&lt;math&gt;T&lt;/math&gt;''', then in '''&lt;math&gt;AB&lt;/math&gt;''' and '''&lt;math&gt;CD&lt;/math&gt;''', '''&lt;math&gt;T&lt;/math&gt;''' and hence '''&lt;math&gt;U&lt;/math&gt;''' remain constant. From the 1st law of thermodynamics:


&lt;math&gt;Q_h = W_{AB} = n\ R\ T_h ln \frac{V_2}{V_1}&lt;/math&gt;
 
&lt;math&gt;Q_c = -W_{CD} = n\ R\ T_c ln \frac{V_3}{V_4}&lt;/math&gt;

*By dividing the equations,

&lt;math&gt;\frac {Q_h}{Q_c} = \frac { T_h ln (\frac {V_2}{V_1})} {T_c ln (\frac{V_3}{V_4})} &lt;/math&gt;

* For '''&lt;math&gt;BC&lt;/math&gt;''' and '''&lt;math&gt;DA&lt;/math&gt;''',

&lt;math&gt;T_h\ V_2^{(\gamma-1)} = T_c\ V_3^{(\gamma-1)}&lt;/math&gt;

&lt;math&gt;T_h V_1^{(\gamma-1)}=T_c\ V_4^{(\gamma-1)}&lt;/math&gt;

* By dividing the equations and taking the &lt;math&gt;(\gamma-1)&lt;/math&gt;th root,

&lt;math&gt;\frac{V_3}{V_4} = \frac{V_2}{V_1}&lt;/math&gt;

*Thus,

&lt;math&gt;\frac{Q_h}{Q_c} = \frac{T_h}{T_c}&lt;/math&gt;

&lt;math&gt;\eta = 1-{\frac{T_c}{T_h}}&lt;/math&gt;

(And this of course applies also to heat pumps and refrigerators. i.e., in their formulas, you can switch the '''&lt;math&gt;Q_h&lt;/math&gt;''' and '''&lt;math&gt;Q_c&lt;/math&gt;''' by '''&lt;math&gt;T_h&lt;/math&gt;''' and '''&lt;math&gt;T_c&lt;/math&gt;''' respectively).

{{BookCat}}</text>
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    <revision>
      <id>2830575</id>
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      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
      </contributor>
      <comment>added Carnot's principles</comment>
      <origin>2830575</origin>
      <model>wikitext</model>
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      <text bytes="4056" sha1="sx9yab8opw2vsgtkv2yyzco1s6a3lsd" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = Carnot cycles
 | previous= The second law of thermodynamics: Kelvin-Planck and Clausius statements
 | next    = Applications of the second law
}}

== The Carnot Engine ==
{{Defbox|name=Carnot's Theorem|value=Carnot’s theorem stated in two principles: The first is that no real (irreversible) engine operating between two heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same two reservoirs. The second is that any two reversible engines operating between the same two reservoirs, will have the same efficiency.|label=Theorem|italic=yes}}
* Carnot made the most efficient heat engine. Net work taken from the Carnot cycle is the largest possible for a given amount of heat supplied.
* Carnot’s theorem can be stated in two principles: 
*# No real (irreversible) engine operating between 2 heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same 2 reservoirs.
*# Any two reversible engines operating between the same two reservoirs will have the same efficiency.
* Carnot used an ideal gas contained in a cylinder with a movable piston at one end. The cylinder walls and the piston are thermally non–conducting.
* The Carnot cycle consists of 2 adiabatic and 2 isothermal processes, all reversible:

'''&lt;math&gt;A \longrightarrow B&lt;/math&gt;'''

Isothermal expansion of a gas placed in thermal contact with a heat reservoir at temperature  '''&lt;math&gt;T_h&lt;/math&gt;'''.

During the process, the gas absorbs heat '''&lt;math&gt;Q_h&lt;/math&gt;''' from the base of the cylinder and does a work '''&lt;math&gt;WAB&lt;/math&gt;''' in raising the piston.

'''&lt;math&gt;B \longrightarrow C&lt;/math&gt;'''

Adiabatic expansion by replacing the base of the cylinder by a thermally non–conducting wall.

During the process, temperature, '''&lt;math&gt;T&lt;/math&gt;''',  falls from '''&lt;math&gt;T_h&lt;/math&gt;''' to '''&lt;math&gt;T_c&lt;/math&gt;''' and the gas does work '''&lt;math&gt;WBC&lt;/math&gt;''' in raising the piston.

'''&lt;math&gt;C \longrightarrow D&lt;/math&gt;'''

Isothermal compression by placing the gas in thermal contact with a heat reservoir at temp. Tc.

During the process, the gas expels heat '''&lt;math&gt;Q_c&lt;/math&gt;''' to the reservoir and the work '''&lt;math&gt;WCD&lt;/math&gt;''' is done on the gas by external agent.

'''&lt;math&gt;D \longrightarrow A&lt;/math&gt;'''

Adiabatic compression by replacing the base of the cylinder by a non–conducting wall.

During the process, temperature, '''&lt;math&gt;T&lt;/math&gt;''', increases from '''&lt;math&gt;T_c&lt;/math&gt;''' to '''&lt;math&gt;T_h&lt;/math&gt;''' and the work '''&lt;math&gt;WDA&lt;/math&gt;''' is done on the gas by external agent.

[[Image:Carnoti.jpg]]
[[File:Máquina_de_Carnot.gif|alt=Máquina de Carnot|left|thumb|500x500px]]
* Net work done in this reversible cyclic process = area enclosed by the path '''&lt;math&gt;ABCDA&lt;/math&gt;'''  = net heat transferred into the system, since '''&lt;math&gt;\Delta U = 0&lt;/math&gt;'''.
* Since the internal '''&lt;math&gt;E&lt;/math&gt;'''. of an ideal gas depends only on absolute '''&lt;math&gt;T&lt;/math&gt;''', then in '''&lt;math&gt;AB&lt;/math&gt;''' and '''&lt;math&gt;CD&lt;/math&gt;''', '''&lt;math&gt;T&lt;/math&gt;''' and hence '''&lt;math&gt;U&lt;/math&gt;''' remain constant. From the 1st law of thermodynamics:


&lt;math&gt;Q_h = W_{AB} = n\ R\ T_h ln \frac{V_2}{V_1}&lt;/math&gt;
 
&lt;math&gt;Q_c = -W_{CD} = n\ R\ T_c ln \frac{V_3}{V_4}&lt;/math&gt;

*By dividing the equations,

&lt;math&gt;\frac {Q_h}{Q_c} = \frac { T_h ln (\frac {V_2}{V_1})} {T_c ln (\frac{V_3}{V_4})} &lt;/math&gt;

* For '''&lt;math&gt;BC&lt;/math&gt;''' and '''&lt;math&gt;DA&lt;/math&gt;''',

&lt;math&gt;T_h\ V_2^{(\gamma-1)} = T_c\ V_3^{(\gamma-1)}&lt;/math&gt;

&lt;math&gt;T_h V_1^{(\gamma-1)}=T_c\ V_4^{(\gamma-1)}&lt;/math&gt;

* By dividing the equations and taking the &lt;math&gt;(\gamma-1)&lt;/math&gt;th root,

&lt;math&gt;\frac{V_3}{V_4} = \frac{V_2}{V_1}&lt;/math&gt;

*Thus,

&lt;math&gt;\frac{Q_h}{Q_c} = \frac{T_h}{T_c}&lt;/math&gt;

&lt;math&gt;\eta = 1-{\frac{T_c}{T_h}}&lt;/math&gt;

(And this of course applies also to heat pumps and refrigerators. i.e., in their formulas, you can switch the '''&lt;math&gt;Q_h&lt;/math&gt;''' and '''&lt;math&gt;Q_c&lt;/math&gt;''' by '''&lt;math&gt;T_h&lt;/math&gt;''' and '''&lt;math&gt;T_c&lt;/math&gt;''' respectively).

{{BookCat}}</text>
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    <revision>
      <id>2830726</id>
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      <comment>/* The Carnot Engine */ updated page for clarity</comment>
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      <text bytes="5182" sha1="j42f5n33g85yl9zld2efnlyr2xk2vqz" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = Carnot cycles
 | previous= The second law of thermodynamics: Kelvin-Planck and Clausius statements
 | next    = Applications of the second law
}}

== The Carnot Engine ==
{{Defbox|name=Carnot's Theorem|value=Carnot’s theorem stated in two principles: The first is that no real (irreversible) engine operating between two heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same two reservoirs. The second is that any two reversible engines operating between the same two reservoirs, will have the same efficiency.|label=Theorem|italic=yes}}Carnot theorized that the most efficient heat engine would be able to extract the maximum amount of net work taken for a given amount of heat supplied and rejected. Carnot’s theorem can be stated in two principles:
# No real (irreversible) engine operating between 2 heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same 2 reservoirs.
# Any two reversible engines operating between the same two reservoirs will have the same efficiency.
It can be shown in two methods that the Carnot engine can be described in terms of the proportional absolute temperature of the reservoirs, in addition to the proportion of the heat absorbed and rejected. 

=== Ideal-gas Carnot cycle ===
Carnot used an ideal gas contained in a cylinder with a movable piston at one end. The cylinder walls and the piston are thermally non–conducting. A few equations to remember before proceeding:

* Isothermal expansion heat input can be written as: &lt;math&gt;Q_H = n R T_H \ln \frac {V_2}{V_1}&lt;/math&gt;
* Isothermal compression heat rejection can be written as: &lt;math&gt;Q_C = n R T_C \ln \frac {V_3}{V_4}&lt;/math&gt;
* Ideal-gas relation tell us that: &lt;math&gt;TV^{\gamma -1} = \text {constant}&lt;/math&gt;
*

The Carnot cycle consists of 2 adiabatic and 2 isothermal processes, all reversible:

'''&lt;math&gt;A \longrightarrow B&lt;/math&gt;'''

Isothermal expansion of a gas placed in thermal contact with a heat reservoir at temperature  '''&lt;math&gt;T_h&lt;/math&gt;'''.

During the process, the gas absorbs heat '''&lt;math&gt;Q_h&lt;/math&gt;''' from the base of the cylinder and does a work '''&lt;math&gt;WAB&lt;/math&gt;''' in raising the piston.

'''&lt;math&gt;B \longrightarrow C&lt;/math&gt;'''

Adiabatic expansion by replacing the base of the cylinder by a thermally non–conducting wall.

During the process, temperature, '''&lt;math&gt;T&lt;/math&gt;''',  falls from '''&lt;math&gt;T_h&lt;/math&gt;''' to '''&lt;math&gt;T_c&lt;/math&gt;''' and the gas does work '''&lt;math&gt;WBC&lt;/math&gt;''' in raising the piston.

'''&lt;math&gt;C \longrightarrow D&lt;/math&gt;'''

Isothermal compression by placing the gas in thermal contact with a heat reservoir at temp. Tc.

During the process, the gas expels heat '''&lt;math&gt;Q_c&lt;/math&gt;''' to the reservoir and the work '''&lt;math&gt;WCD&lt;/math&gt;''' is done on the gas by external agent.

'''&lt;math&gt;D \longrightarrow A&lt;/math&gt;'''

Adiabatic compression by replacing the base of the cylinder by a non–conducting wall.

During the process, temperature, '''&lt;math&gt;T&lt;/math&gt;''', increases from '''&lt;math&gt;T_c&lt;/math&gt;''' to '''&lt;math&gt;T_h&lt;/math&gt;''' and the work '''&lt;math&gt;WDA&lt;/math&gt;''' is done on the gas by external agent.

[[Image:Carnoti.jpg]]
[[File:Máquina_de_Carnot.gif|alt=Máquina de Carnot|left|thumb|500x500px]]
* Net work done in this reversible cyclic process = area enclosed by the path '''&lt;math&gt;ABCDA&lt;/math&gt;'''  = net heat transferred into the system, since '''&lt;math&gt;\Delta U = 0&lt;/math&gt;'''.
* Since the internal '''&lt;math&gt;E&lt;/math&gt;'''. of an ideal gas depends only on absolute '''&lt;math&gt;T&lt;/math&gt;''', then in '''&lt;math&gt;AB&lt;/math&gt;''' and '''&lt;math&gt;CD&lt;/math&gt;''', '''&lt;math&gt;T&lt;/math&gt;''' and hence '''&lt;math&gt;U&lt;/math&gt;''' remain constant. From the 1st law of thermodynamics:


&lt;math&gt;Q_h = W_{AB} = n\ R\ T_h ln \frac{V_2}{V_1}&lt;/math&gt;
 
&lt;math&gt;Q_c = -W_{CD} = n\ R\ T_c ln \frac{V_3}{V_4}&lt;/math&gt;

*By dividing the equations,

&lt;math&gt;\frac {Q_h}{Q_c} = \frac { T_h ln (\frac {V_2}{V_1})} {T_c ln (\frac{V_3}{V_4})} &lt;/math&gt;

* For '''&lt;math&gt;BC&lt;/math&gt;''' and '''&lt;math&gt;DA&lt;/math&gt;''',

&lt;math&gt;T_h\ V_2^{(\gamma-1)} = T_c\ V_3^{(\gamma-1)}&lt;/math&gt;

&lt;math&gt;T_h V_1^{(\gamma-1)}=T_c\ V_4^{(\gamma-1)}&lt;/math&gt;

* By dividing the equations and taking the &lt;math&gt;(\gamma-1)&lt;/math&gt;th root,

&lt;math&gt;\frac{V_3}{V_4} = \frac{V_2}{V_1}&lt;/math&gt;

*Thus,

&lt;math&gt;\frac{Q_h}{Q_c} = \frac{T_h}{T_c}&lt;/math&gt;

&lt;math&gt;\eta = 1-{\frac{T_c}{T_h}}&lt;/math&gt;

(And this of course applies also to heat pumps and refrigerators. i.e., in their formulas, you can switch the '''&lt;math&gt;Q_h&lt;/math&gt;''' and '''&lt;math&gt;Q_c&lt;/math&gt;''' by '''&lt;math&gt;T_h&lt;/math&gt;''' and '''&lt;math&gt;T_c&lt;/math&gt;''' respectively).

=== Three reversible heat engines ===
Three reversible heat engines can be used to show that the thermal efficiency of a Carnot heat engine is a function of the reservoir temperatures only. Absolute temperatures can be defined so that their ratios equal exactly the heat-transfer ratios of reversible heat engines operating between them. 

Put two reversible Carnot engines in series and if they between two reservoirs, then they will have the same efficiency as one Carnot engine (reversible) operating between the same two reservoirs.{{BookCat}}</text>
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    <title>Introduction to thermodynamics/Refrigerator and heat pump</title>
    <ns>0</ns>
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        <id>2918363</id>
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      <comment>added example</comment>
      <origin>2830567</origin>
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      <text bytes="3710" sha1="9gprskj0qxs2f9pblh3100tqf9dv261" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = Refrigerator and heat pump
 | previous= Applications of the first law
 | next    = The second law of thermodynamics: Kelvin-Planck and Clausius statements
}}

== Heat pumps ==

Heat pumps are devices that convert work into useful thermal energy transfer (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;), which are used to heat/cool homes. During the cycle, heat, (&lt;math display=&quot;inline&quot;&gt;{Q_c}&lt;/math&gt;), is absorbed from a source at low &lt;math display=&quot;inline&quot;&gt;\scriptstyle {T}&lt;/math&gt; (e.g. outside air or food) by a circulating fluid, usually a refrigerant. 

Work is done on the engine by a compressor. Heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is expelled to a source at higher &lt;math display=&quot;inline&quot;&gt;\scriptstyle {T}&lt;/math&gt; (e.g. room).

Since it is a cyclic process: &lt;math display=&quot;inline&quot;&gt;\scriptstyle {\Delta U=0} &lt;/math&gt;. Thus, 

&lt;math&gt;W = Q_{\text{total}} \longrightarrow W = Q_h - Q_c &lt;/math&gt; 

&lt;math display=&quot;inline&quot;&gt;COP = \text {coefficient of performance} = \text {ratio of heat transferred to work required to transfer the heat}&lt;/math&gt;

&lt;math&gt;COP = \frac{Q_h}{W} = \frac{Q_h}{Q_h - Q_c} &lt;/math&gt;

&lt;math&gt;COP &lt;/math&gt; can be much larger than 100% (denominator &lt; 1).

For example, imagine that a heat pump is used to heat a house at 20 °C in the winter. During a particularly cold day, the outside temperature drops to -10 °C, and the house loses 120,000 kJ/hour. The coefficient of performance (COP) of the heat pump is 2.5. We are interested in finding a) the required work input for the heat pump, b) the required heat input from the surrounding and c) the benefit of a heat pump relative to a electrical resistance heater on the electrical bill.

First we must make the assumption that the cycle exists in steady-state conditions. Next define, the systems; the first is the house and the second is the system surrounding the heat-pump. For steady-state conditions in the system comprising the house, the the total change in energy must be zero. This means that the energy leaving the house must be equal to the energy entering the house, or that 120,000 kJ/hour must enter the house in order to balance the 120,000 kJ/hour that leaves the house. 

As COP_heat,pump is 2.5, we can find the required work-input of the compressor:

== Refrigerators ==

{{Blockquote|text=It is impossible to use a cyclic process to transfer heat from a colder to a hotter body without doing work on the system.|author=Clausius form of the 2nd law of thermodynamics}}

In other words heat will not flow spontaneously from a cold to a hot object.

Refrigerators are devices that convert work into thermal energy, &lt;math display=&quot;inline&quot;&gt;E&lt;/math&gt;, (&lt;math display=&quot;inline&quot;&gt;{Q_c}&lt;/math&gt;).

During the cycle,

Heat (&lt;math display=&quot;inline&quot;&gt;{Q_c}&lt;/math&gt;) is absorbed from a source at low &lt;math&gt;T &lt;/math&gt; (e.g. outside air or food) by a circulating fluid.

Work is done on the engine by a compressor.

Heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is expelled to a source at higher &lt;math&gt;T &lt;/math&gt; (e.g. room ambient temperature).

Since it is a cyclic process, &lt;math&gt;\Delta U = 0 &lt;/math&gt;. Thus, &lt;math&gt;W = Q_{total} \Longrightarrow W = Q_h - Q_c &lt;/math&gt; 

&lt;math display=&quot;inline&quot;&gt;COP = \text {coefficient of performance} = \text {ratio of heat transferred to work required to transfer the heat}&lt;/math&gt;

&lt;math&gt;COP = \frac {Qc}{W} = \frac {Qc}{(Qh - Qc)} &lt;/math&gt;

&lt;math&gt;COP &lt;/math&gt; can be much larger than 100% (denominator &lt; 1).

A perfect refrigerator would transfer heat from a colder body to a hotter body without doing any work. From 2nd law of thermodynamics, this is impossible. For example the &lt;math&gt;COP &lt;/math&gt; of a refrigerator is around 5 or 6.

{{BookCat}}</text>
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        <username>IanVG</username>
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      <comment>finished problem</comment>
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      <text bytes="4755" sha1="sesgpxpog4mllqeb6p1km711wa0vqse" xml:space="preserve">{{Chapter
 | title   = Introduction to thermodynamics
 | number  = 8
 | lesson  = Refrigerator and heat pump
 | previous= Applications of the first law
 | next    = The second law of thermodynamics: Kelvin-Planck and Clausius statements
}}

== Heat pumps ==

Heat pumps are devices that convert work into useful thermal energy transfer (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;), which are used to heat/cool homes. During the cycle, heat, (&lt;math display=&quot;inline&quot;&gt;{Q_c}&lt;/math&gt;), is absorbed from a source at low &lt;math display=&quot;inline&quot;&gt;\scriptstyle {T}&lt;/math&gt; (e.g. outside air or food) by a circulating fluid, usually a refrigerant. 

Work is done on the engine by a compressor. Heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is expelled to a source at higher &lt;math display=&quot;inline&quot;&gt;\scriptstyle {T}&lt;/math&gt; (e.g. room).

Since it is a cyclic process: &lt;math display=&quot;inline&quot;&gt;\scriptstyle {\Delta U=0} &lt;/math&gt;. Thus, 

&lt;math&gt;W = Q_{\text{total}} \longrightarrow W = Q_h - Q_c &lt;/math&gt; 

&lt;math display=&quot;inline&quot;&gt;COP = \text {coefficient of performance} = \text {ratio of heat transferred to work required to transfer the heat}&lt;/math&gt;

&lt;math&gt;COP = \frac{Q_h}{W} = \frac{Q_h}{Q_h - Q_c} &lt;/math&gt;

&lt;math&gt;COP&lt;/math&gt; can be much larger than 100% (denominator &lt; 1).

For example, imagine that a heat pump is used to heat a house at 20 °C in the winter. During a particularly cold day, the outside temperature drops to -10 °C, and the house loses 120,000 kJ/hour. The coefficient of performance (COP) of the heat pump is 2.5. We are interested in finding a) the required work input for the heat pump, b) the required heat input from the surrounding and c) the benefit of a heat pump relative to a electrical resistance heater on the electrical bill.

First we must make the assumption that the cycle exists in steady-state conditions. Next define, the systems; the first is the house and the second is the system surrounding the heat-pump. For steady-state conditions in the system comprising the house, the the total change in energy must be zero. This means that the energy leaving the house must be equal to the energy entering the house, or that 120,000 kJ/hour must enter the house in order to balance the 120,000 kJ/hour that leaves the house. 

As COP_heat,pump is 2.5, we can find the required work-input of the compressor:

&lt;math&gt;COP_{HP} = \frac {\text {desired output}}{\text {required input}} = \frac {\dot Q_{supplied}}{\dot W_{in}} \Longrightarrow \dot W_{in} = \frac { 120,000 \, kJ/hr }{2.5} = 48,000 \, kJ/hr &lt;/math&gt;

Applying the first law to the heat pump we find that:

&lt;math&gt;\dot Q_{in} + \dot W_{in} = \dot Q_{supplied} &lt;/math&gt;

Therefore, the difference between the heat entering the house and the work supplied to the compressor, is the amount of heat that the heat pump takes from the surroundings, or 120,000 kJ/hr - 48,000 kJ/hr = 72,000 kJ/hr = \dot Q_{in}. This means that the heat pump only 48,000 kJ/hr of input power in order to supply the house with 120,000 kJ/hr. If we were to use an electrical resistance heater, 120,000 kJ/hr of electrical energy would need to be supplied in order to maintain the desired interior temperature in the house.This means that the heat pump saves us 72,000 kJ/hr compared to an electrical heater. This demonstrates that heat pumps have a clear advantage over electrical resistance heaters for heating buildings.

== Refrigerators ==

{{Blockquote|text=It is impossible to use a cyclic process to transfer heat from a colder to a hotter body without doing work on the system.|author=Clausius form of the 2nd law of thermodynamics}}

In other words heat will not flow spontaneously from a cold to a hot object.

Refrigerators are devices that convert work into thermal energy, &lt;math display=&quot;inline&quot;&gt;E&lt;/math&gt;, (&lt;math display=&quot;inline&quot;&gt;{Q_c}&lt;/math&gt;).

During the cycle,

Heat (&lt;math display=&quot;inline&quot;&gt;{Q_c}&lt;/math&gt;) is absorbed from a source at low &lt;math&gt;T &lt;/math&gt; (e.g. outside air or food) by a circulating fluid.

Work is done on the engine by a compressor.

Heat (&lt;math display=&quot;inline&quot;&gt;Q_h&lt;/math&gt;) is expelled to a source at higher &lt;math&gt;T &lt;/math&gt; (e.g. room ambient temperature).

Since it is a cyclic process, &lt;math&gt;\Delta U = 0 &lt;/math&gt;. Thus, &lt;math&gt;W = Q_{total} \Longrightarrow W = Q_h - Q_c &lt;/math&gt; 

&lt;math display=&quot;inline&quot;&gt;COP = \text {coefficient of performance} = \text {ratio of heat transferred to work required to transfer the heat}&lt;/math&gt;

&lt;math&gt;COP = \frac {Qc}{W} = \frac {Qc}{(Qh - Qc)} &lt;/math&gt;

&lt;math&gt;COP &lt;/math&gt; can be much larger than 100% (denominator &lt; 1).

A perfect refrigerator would transfer heat from a colder body to a hotter body without doing any work. From 2nd law of thermodynamics, this is impossible. For example the &lt;math&gt;COP &lt;/math&gt; of a refrigerator is around 5 or 6.

{{BookCat}}</text>
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    <title>OpenStax Business Ethics</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <text bytes="671" sha1="3qwcbj4ganeyxgxcklyl4nfyqnuqgrb" xml:space="preserve">See also [[OpenStax]]

== OpenStax Business Ethics ==
''Business Ethics'' is designed to meet the scope and sequence requirements of the single-semester business ethics course. This title includes innovative features designed to enhance student learning, including case studies, application scenarios, and links to video interviews with executives, all of which help instill in students a sense of ethical awareness and responsibility.

[https://openstax.org/details/books/business-ethics OpenStax Business Ethics] (original content)

[https://audileo.com/audiobooks/openstax/business-ethics/ OpenStax Business Ethics audiobook]

[[Category:Ethics]]
[[Category:OpenStax]]</text>
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    <title>OpenStax Introduction to Business</title>
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      <text bytes="1084" sha1="0ae72do6x0cpqiu5conqe4ue64990w6" xml:space="preserve">See also [[OpenStax]]

See also [[OpenStax Introduction to Business 2e]]

== OpenStax Introduction to Business ==
Introduction to Business covers the scope and sequence of most introductory business courses. The book provides detailed explanations in the context of core themes such as customer satisfaction, ethics, entrepreneurship, global business, and managing change. Introduction to Business includes hundreds of current business examples from a range of industries and geographic locations, which feature a variety of individuals. The outcome is a balanced approach to the theory and application of business concepts, with attention to the knowledge and skills necessary for student success in this course and beyond.

[https://openstax.org/details/books/introduction-business OpenStax Introduction to Business] (original content)

[https://audileo.com/audiobooks/openstax/introduction-to-business/ OpenStax Introduction to Business audiobook] (official audiobook, [https://www.youtube.com/watch?v=owZJXiKbNFQ chapter 1 on YouTube])

[[Category:Business]]
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      <text bytes="1832" sha1="qym3osyg3jf8j9ga89coczz5bwkk7v5" xml:space="preserve">See also [[OpenStax]]

See also [[OpenStax Introduction to Business]]

== OpenStax Introduction to Business 2e ==
Introduction to Business 2e provides a strong, multidimensional foundation in the concepts, skills, and real-world examples of contemporary business. Welcoming and relevant narratives support students as they engage in core themes such as decision making, ethics, customer satisfaction, entrepreneurship, global business, and managing technological and societal changes. The material includes hundreds of current business examples from a range of industries and geographic locations. Extensive illustrations, case studies, review materials, and assessments offer instructors and learners the tools needed to develop a deep, practical understanding. The outcome is a balanced approach to the theory and application of business concepts, with attention to the knowledge and skills necessary for success in further studies and careers.

The Second Edition was written with extensive input from faculty, industry experts, and students. Revisions focus on currency, emerging technologies, changing business practices, and evolving legal and ethical dynamics. The authors emphasize course-wide themes, including the impacts and implications of artificial intelligence, the changing nature of the workplace, and new approaches to team dynamics. A detailed transition guide is available in the instructor resources.

[https://openstax.org/details/books/introduction-business-2e OpenStax Introduction to Business 2e] (original content)

[https://audileo.com/audiobooks/openstax/introduction-to-business-2e/ OpenStax Introduction to Business 2e audiobook] (official audiobook)

[https://www.youtube.com/watch?v=ylt-Wc3UD7A OpenStax Introduction to Business 2e] (chapter 1 on YouTube)

[[Category:Business]]
[[Category:OpenStax]]</text>
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    <title>OpenStax Introduction to Sociology 3e</title>
    <ns>0</ns>
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== OpenStax Introduction to Sociology 3e ==
Introduction to Sociology 3e aligns to the topics and objectives of many introductory sociology courses. It is arranged in a manner that provides foundational sociological theories and contexts, then progresses through various aspects of human and societal interactions. The new edition is focused on driving meaningful and memorable learning experiences related to critical thinking about society and culture. The text includes comprehensive coverage of core concepts, discussions and data relevant to a diverse audience, and features that draw learners into the discipline in powerful and personal ways. Overall, Introduction to Sociology 3e aims to center the course and discipline as crucial elements for understanding relationships, society, and civic engagement; the authors seek to lay the foundation for students to apply what they learn throughout their lives and careers.


[https://openstax.org/details/books/introduction-sociology-3e OpenStax Introduction to Sociology 3e] (original content)

[https://audileo.com/audiobooks/openstax/introduction-to-sociology-3e/ OpenStax Introduction to Sociology 3e Audiobook] (full audiobook)

[[Category:Sociology]]
[[Category:OpenStax]]</text>
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    <title>OpenStax Introduction to Anthropology</title>
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      <text bytes="1529" sha1="jxgh3ar1qv1x1is9gijrgmfejwqvrjd" xml:space="preserve">See also [[OpenStax]]

== OpenStax Introduction to Anthropology ==
Designed to meet the scope and sequence of your course, OpenStax ''Introduction to Anthropology'' is a four-field text integrating diverse voices, engaging field activities, and meaningful themes like Indigenous experiences and social inequality to engage students and enrich learning. The text showcases the historical context of the discipline, with a strong focus on anthropology as a living and evolving field.  There is significant discussion of recent efforts to make the field more diverse—in its practitioners, in the questions it asks, and in the applications of anthropological research to address contemporary challenges.  In addressing social inequality, the text drives readers to consider the rise and impact of social inequalities based on forms of identity and difference (such as gender, ethnicity, race, and class) as well as oppression and discrimination. The contributors to and dangers of socioeconomic inequality are fully addressed, and the role of inequality in social dysfunction, disruption, and change is noted.

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[https://www.youtube.com/watch?v=JhYpOZD1Kyo OpenStax Introduction to Anthropology] (chapter 1 on YouTube)

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== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Principles of&lt;/big&gt; &lt;big&gt;Economics 3e&lt;/big&gt; ==

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See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;College Success Concise&lt;/big&gt; ==

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== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Organizational Behavior&lt;/big&gt; ==

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      <text bytes="36753" sha1="o4rluhcnfjfgavi8awmraammdgn203j" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*The previous section ended on an open question: whether dreams actively regulate emotion or passively mirror waking life. Four theories answer it differently: three claim dreaming does something while one claims it does not, and that is the position that the other three have to beat. They cannot be separated by dream content, because all four expect distressing dreams after distressing events. They differ in what should be true the next morning (see Table 1). [[w:Nightmare|Nightmares]] make the clearest test among the three function theories, because the same dream counts as either the system  working for one theory and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! Core claim !! Predicts that after a distressing dream, next-day mood will be... !! What it would say about Natalie
|-
| '''Continuity''' || Dreams reflect waking concerns but do not act on them || Unchanged. The dream mirrors mood without moving it || The dream is an accurate reflection of the breakup and nothing more
|-
| '''Threat simulation''' || Dreaming rehearses detecting and avoiding threat || No better, possibly worse. The payoff is preparedness || The dream is training her for abandonment, not calming her about it 
|-
| '''Fear extinction''' || Fear memories are replayed in safe new contexts, building a competing safe memory || Better if extinction succeeds. Unchanged or worse if it fails || Her dream failed replay: fear is rehearsed but no safe version is built. 
|-
| '''Mood regulation''' || Dream emotion shifts from negative toward positive overnight, discharging its intensity || Better: less negative mood, weaker reactivity, better regulation || The dream should be fading week by week, and it is not
|-
|}

=== The continuity hypothesis ===

* Dreams reflect waking concerns without acting on them, so nothing more is needed to explain why Natalie dreams about being left behind (Domhoff, 2017). 
* This is the position that the other theories have to beat rather than fourth option sitting alongside them: showing that a dream is meaningful is not the same as showing that is does something (Domhoff, 2019).
* Even continuity is contested. Judges have repeatedly failed to match dream reports to what the dreamer was concerned about the day before (Domhoff, 2017). Others argue the effect is real but selective, with emotionally salient concerns incorporated far more often than trivial ones (Schredl, 2017; Malinowski &amp; Horton, 2015).
* Applied to Natalie, continuity predicts the dream keeps reoccurring and her mood keeps tracking it, with no improvement produced by the dreaming itself.

===Threat simulation ===

* Here the dream adds rehearsal. Dreaming simulates threatening events and practices detecting and avoiding them. The payoff is preparedness, not comfort, so nightmares show the system working rather than failing (Revonsuo, 2000).
* The supporting claim is that most dream emotion is negative and aggression is the most common social interaction in dreams (Revonsuo, 2000). However, that estimate depends on who does the rating, since dreamers rate their own dreams as mostly positive (Sikka et al., 2017).
* Against it: among recurrent dreams, 34% contained no threat at all and fewer than 20% resolved one, so the theory's most specific predictions were the ones that failed (Desjardins &amp; Zadra, 2006). It also requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* Here the dream adds recombination: frightening memories are replayed in new, non-fearful settings (Nielsen &amp; Levin, 2007). [[w:Extinction (psychology)|Extinction]] doesn't erase the fear, it builds a competing safe memory alongside it, which is why the fear returns if that new memory isn't maintained (Nielsen &amp; Levin, 2007).
* On this account nightmares are a breakdown of this process, as the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* Disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (PTSD), which gives the model a clinical test case (Pace-Schott et al., 2015).
* Cathartic dreams, which move from danger to relief, are less common in nightmare disorder than in healthy sleepers, and were the only dream type that tracked falling depression scores across therapy (Perogamvros et al., 2025).
* The main criticism is that the model treats bad dreams as adaptive and nightmares as maladaptive while attributing both to a single process, and it has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007).
* This is the theory tested most directly in ''Using dreams to improve coping'', because sleep interventions manipulate what gets replayed.

===Mood regulation ===

* Here the dream adds desensitisation. Dream emotion is proposed to shift from negative toward positive across the night, so its charge is reduced by morning (Cartwright et al., 1998). This is the headline claim, and the one closest to the popular idea that dreams work through problems, and it is tested in ''Dream emotion and waking life.''
* Dreamer's own ratings of the emotional tone of their day-to-day dreams have been read as inconsistent with a mood regulation function (Barbeau et al., 2022).
* The supporting evidence is real but indirect: people who report more fear in dreams show weaker fear-related brain responses when awake (Sterpenich et al., 2020). That is a correlation across people at a single time point, so it can't establish direction. 
* The theory makes three testable predictions for the morning after a negative dream: mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be better (Sikka et al., 2022).
* Applied to Natalie, it predicts that a dream should be losing intensity over time, and hers is not.

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
Baber, G. R., Hamilton, N. A., Girard, J. M., Sikka, P., Watts, A., Jun, D., Gratton, M. K. P., Cohen, J. M., Quesada, A. K., Nichols, E. C., &amp; Cunningham, T. J. (2026). Testing affect regulation theories of dreaming. ''Sleep, 49''(5). https://doi.org/10.1093/sleep/zsag046 

Barbeau, K., Turpin, C., Lafrenière, A., Campbell, E., &amp; De Koninck, J. (2022). Dreamers' evaluation of the emotional valence of their day-to-day dreams is indicative of some mood regulation function. ''Frontiers in Behavioral Neuroscience, 16'', 947396. https://doi.org/10.3389/fnbeh.2022.947396

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! Core claim !! Predicts that after a distressing dream, next-day mood will be... !! What it would say about Natalie
|-
| '''Continuity''' || Dreams reflect waking concerns but do not act on them || Unchanged. The dream mirrors mood without moving it || The dream is an accurate reflection of the breakup and nothing more
|-
| '''Threat simulation''' || Dreaming rehearses detecting and avoiding threat || No better, possibly worse. The payoff is preparedness || The dream is training her for abandonment, not calming her about it 
|-
| '''Fear extinction''' || Fear memories are replayed in safe new contexts, building a competing safe memory || Better if extinction succeeds. Unchanged or worse if it fails || Her dream failed replay: fear is rehearsed but no safe version is built. 
|-
| '''Mood regulation''' || Dream emotion shifts from negative toward positive overnight, discharging its intensity || Better: less negative mood, weaker reactivity, better regulation || The dream should be fading week by week, and it is not
|-
|}

=== The continuity hypothesis ===

* Dreams reflect waking concerns without acting on them, so nothing more is needed to explain why Natalie dreams about being left behind (Domhoff, 2017). 
* This is the position that the other theories have to beat rather than fourth option sitting alongside them: showing that a dream is meaningful is not the same as showing that is does something (Domhoff, 2019).
* Even continuity is contested. Judges have repeatedly failed to match dream reports to what the dreamer was concerned about the day before (Domhoff, 2017). Others argue the effect is real but selective, with emotionally salient concerns incorporated far more often than trivial ones (Schredl, 2017; Malinowski &amp; Horton, 2015).
* Applied to Natalie, continuity predicts the dream keeps reoccurring and her mood keeps tracking it, with no improvement produced by the dreaming itself.

===Threat simulation ===

* Here the dream adds rehearsal. Dreaming simulates threatening events and practices detecting and avoiding them. The payoff is preparedness, not comfort, so nightmares show the system working rather than failing (Revonsuo, 2000).
* The supporting claim is that most dream emotion is negative and aggression is the most common social interaction in dreams (Revonsuo, 2000). However, that estimate depends on who does the rating, since dreamers rate their own dreams as mostly positive (Sikka et al., 2017).
* Against it: among recurrent dreams, 34% contained no threat at all and fewer than 20% resolved one, so the theory's most specific predictions were the ones that failed (Desjardins &amp; Zadra, 2006). It also requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* Here the dream adds recombination: frightening memories are replayed in new, non-fearful settings (Nielsen &amp; Levin, 2007). [[w:Extinction (psychology)|Extinction]] doesn't erase the fear, it builds a competing safe memory alongside it, which is why the fear returns if that new memory isn't maintained (Nielsen &amp; Levin, 2007).
* On this account nightmares are a breakdown of this process, as the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* Disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (PTSD), which gives the model a clinical test case (Pace-Schott et al., 2015).
* Cathartic dreams, which move from danger to relief, are less common in nightmare disorder than in healthy sleepers, and were the only dream type that tracked falling depression scores across therapy (Perogamvros et al., 2025).
* The main criticism is that the model treats bad dreams as adaptive and nightmares as maladaptive while attributing both to a single process, and it has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007).
* This is the theory tested most directly in ''Using dreams to improve coping'', because sleep interventions manipulate what gets replayed.

===Mood regulation ===

* Here the dream adds desensitisation. Dream emotion is proposed to shift from negative toward positive across the night, so its charge is reduced by morning (Cartwright et al., 1998). This is the headline claim, and the one closest to the popular idea that dreams work through problems, and it is tested in ''Dream emotion and waking life.''
* Dreamer's own ratings of the emotional tone of their day-to-day dreams have been read as inconsistent with a mood regulation function (Barbeau et al., 2022).
* The supporting evidence is real but indirect: people who report more fear in dreams show weaker fear-related brain responses when awake (Sterpenich et al., 2020). That is a correlation across people at a single time point, so it can't establish direction. 
* The theory makes three testable predictions for the morning after a negative dream: mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be better (Sikka et al., 2022).
* Applied to Natalie, it predicts that a dream should be losing intensity over time, and hers is not.

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Harb, G. C., Cook, J. M., Phelps, A. J., Gehrman, P. R., Forbes, D., Localio, R., Harpaz-Rotem, I., Gur, R. C., &amp; Ross, R. J. (2019). Randomized controlled trial of imagery rehearsal for posttraumatic nightmares in combat veterans. ''Journal of Clinical Sleep Medicine, 15''(5), 757–767. https://doi.org/10.5664/jcsm.7770

Kempe, S., Köpp, W., &amp; Wittmann, L. (2024). Personality functioning improvement during psychotherapy is associated with an enhanced capacity for affect regulation in dreams: A preliminary study. ''Brain Sciences, 14''(5), 489. https://doi.org/10.3390/brainsci14050489

Lara-Carrasco, J., Nielsen, T. A., Solomonova, E., Levrier, K., &amp; Popova, A. (2009). Overnight emotional adaptation to negative stimuli is altered by REM sleep deprivation and is correlated with intervening dream emotions. ''Journal of Sleep Research, 18''(2), 178–187. https://doi.org/10.1111/j.1365-2869.2008.00709.x

Lipinska, G., Austin, H., Moonsamy, J. R., Henry, M., Lewis, R., Baldwin, D. S., Thomas, K. G. F., &amp; Stuart, B. (2022). Preferential consolidation of emotional reactivity during sleep: A systematic review and meta-analysis. ''Frontiers in Behavioral Neuroscience, 16'', 976047. https://doi.org/10.3389/fnbeh.2022.976047

Lipinska, G., Stuart, B., Thomas, K. G. F., Baldwin, D. S., &amp; Bolinger, E. (2019). Preferential consolidation of emotional memory during sleep: A meta-analysis. ''Frontiers in Psychology, 10'', 1014. https://doi.org/10.3389/fpsyg.2019.01014

Malinowski, J. E., &amp; Horton, C. L. (2015). Metaphor and hyperassociativity: The imagination mechanisms behind emotion assimilation in sleep and dreaming. ''Frontiers in Psychology, 6'', 1132. https://doi.org/10.3389/fpsyg.2015.01132

Mendoza Alvarez, M., Balthasar, Y., Verbraecken, J., Claes, L., van Someren, E., van Marle, H. J. F., Vandekerckhove, M., &amp; De Picker, L. (2024). Systematic review: REM sleep, dysphoric dreams and nightmares as transdiagnostic features of psychiatric disorders with emotion dysregulation – Clinical implications. ''Sleep Medicine, 127'', 1–15. https://doi.org/10.1016/j.sleep.2024.12.037

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Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== The continuity hypothesis ===

* Dreams reflect waking concerns without acting on them, so nothing more is needed to explain why Natalie dreams about being left behind (Domhoff, 2017). 
* This is the position that the other theories have to beat rather than fourth option sitting alongside them: showing that a dream is meaningful is not the same as showing that is does something (Domhoff, 2019).
* Even continuity is contested. Judges have repeatedly failed to match dream reports to what the dreamer was concerned about the day before (Domhoff, 2017). Others argue the effect is real but selective, with emotionally salient concerns incorporated far more often than trivial ones (Schredl, 2017; Malinowski &amp; Horton, 2015).
* Applied to Natalie, continuity predicts the dream keeps reoccurring and her mood keeps tracking it, with no improvement produced by the dreaming itself.

===Threat simulation ===

* Here the dream adds rehearsal. Dreaming simulates threatening events and practices detecting and avoiding them. The payoff is preparedness, not comfort, so nightmares show the system working rather than failing (Revonsuo, 2000).
* The supporting claim is that most dream emotion is negative and aggression is the most common social interaction in dreams (Revonsuo, 2000). However, that estimate depends on who does the rating, since dreamers rate their own dreams as mostly positive (Sikka et al., 2017).
* Against it: among recurrent dreams, 34% contained no threat at all and fewer than 20% resolved one, so the theory's most specific predictions were the ones that failed (Desjardins &amp; Zadra, 2006). It also requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* Here the dream adds recombination: frightening memories are replayed in new, non-fearful settings (Nielsen &amp; Levin, 2007). [[w:Extinction (psychology)|Extinction]] doesn't erase the fear, it builds a competing safe memory alongside it, which is why the fear returns if that new memory isn't maintained (Nielsen &amp; Levin, 2007).
* On this account nightmares are a breakdown of this process, as the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* Disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (PTSD), which gives the model a clinical test case (Pace-Schott et al., 2015).
* Cathartic dreams, which move from danger to relief, are less common in nightmare disorder than in healthy sleepers, and were the only dream type that tracked falling depression scores across therapy (Perogamvros et al., 2025).
* The main criticism is that the model treats bad dreams as adaptive and nightmares as maladaptive while attributing both to a single process, and it has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007).
* This is the theory tested most directly in ''Using dreams to improve coping'', because sleep interventions manipulate what gets replayed.

===Mood regulation ===

* Here the dream adds desensitisation. Dream emotion is proposed to shift from negative toward positive across the night, so its charge is reduced by morning (Cartwright et al., 1998). This is the headline claim, and the one closest to the popular idea that dreams work through problems, and it is tested in ''Dream emotion and waking life.''
* Dreamer's own ratings of the emotional tone of their day-to-day dreams have been read as inconsistent with a mood regulation function (Barbeau et al., 2022).
* The supporting evidence is real but indirect: people who report more fear in dreams show weaker fear-related brain responses when awake (Sterpenich et al., 2020). That is a correlation across people at a single time point, so it can't establish direction. 
* The theory makes three testable predictions for the morning after a negative dream: mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be better (Sikka et al., 2022).
* Applied to Natalie, it predicts that a dream should be losing intensity over time, and hers is not.

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
*

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="36993" sha1="onjh3ijkg1i6n3ag7h3y9yisthgpf55" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
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==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== The continuity hypothesis ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* Here the dream adds rehearsal. Dreaming simulates threatening events and practices detecting and avoiding them. The payoff is preparedness, not comfort, so nightmares show the system working rather than failing (Revonsuo, 2000).
* The supporting claim is that most dream emotion is negative and aggression is the most common social interaction in dreams (Revonsuo, 2000). However, that estimate depends on who does the rating, since dreamers rate their own dreams as mostly positive (Sikka et al., 2017).
* Against it: among recurrent dreams, 34% contained no threat at all and fewer than 20% resolved one, so the theory's most specific predictions were the ones that failed (Desjardins &amp; Zadra, 2006). It also requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* Here the dream adds recombination: frightening memories are replayed in new, non-fearful settings (Nielsen &amp; Levin, 2007). [[w:Extinction (psychology)|Extinction]] doesn't erase the fear, it builds a competing safe memory alongside it, which is why the fear returns if that new memory isn't maintained (Nielsen &amp; Levin, 2007).
* On this account nightmares are a breakdown of this process, as the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* Disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (PTSD), which gives the model a clinical test case (Pace-Schott et al., 2015).
* Cathartic dreams, which move from danger to relief, are less common in nightmare disorder than in healthy sleepers, and were the only dream type that tracked falling depression scores across therapy (Perogamvros et al., 2025).
* The main criticism is that the model treats bad dreams as adaptive and nightmares as maladaptive while attributing both to a single process, and it has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007).
* This is the theory tested most directly in ''Using dreams to improve coping'', because sleep interventions manipulate what gets replayed.

===Mood regulation ===

* Here the dream adds desensitisation. Dream emotion is proposed to shift from negative toward positive across the night, so its charge is reduced by morning (Cartwright et al., 1998). This is the headline claim, and the one closest to the popular idea that dreams work through problems, and it is tested in ''Dream emotion and waking life.''
* Dreamer's own ratings of the emotional tone of their day-to-day dreams have been read as inconsistent with a mood regulation function (Barbeau et al., 2022).
* The supporting evidence is real but indirect: people who report more fear in dreams show weaker fear-related brain responses when awake (Sterpenich et al., 2020). That is a correlation across people at a single time point, so it can't establish direction. 
* The theory makes three testable predictions for the morning after a negative dream: mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be better (Sikka et al., 2022).
* Applied to Natalie, it predicts that a dream should be losing intensity over time, and hers is not.

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
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===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
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==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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Palmieri, J., Elce, V., &amp; Schönauer, M. (2025). Nightly dynamics of emotional content in dreams. ''Neuropsychologia, 218''(1), 109248. https://doi.org/10.1016/j.neuropsychologia.2025.109248 

Peltz, J., Rogge, R. D., &amp; Zeitzer, J. M. (2026). The transactional role of rapid eye movement sleep in undergraduates' day-to-day positive and negative mood. ''Journal of Sleep Research'', e70373. https://doi.org/10.1111/jsr.70373 

Perogamvros, L., Rochas, V., Beau, J. B., Sterpenich, V., &amp; Bayer, L. (2025). The cathartic dream: Using a large language model to study a new type of functional dream in healthy and clinical populations. ''Journal of Sleep Research, 34''(6), e70001. https://doi.org/10.1111/jsr.70001

Pesonen, A.-K., Koskinen, M.-K., Vuorenhela, N., Halonen, R., Mäkituuri, S., Selin, M., Luokkala, S., Suutari, A., &amp; Hovatta, I. (2024). The effect of REM-sleep disruption on affective processing: A systematic review of human and animal experimental studies. ''Neuroscience &amp; Biobehavioral Reviews, 162'', 105714. https://doi.org/10.1016/j.neubiorev.2024.105714

Revonsuo, A. (2000). The reinterpretation of dreams: An evolutionary hypothesis of the function of dreaming. ''Behavioral and Brain Sciences, 23''(6), 877–901. https://doi.org/10.1017/S0140525X00004015

Rousseau, A., &amp; Belleville, G. (2018). The mechanisms of action underlying the efficacy of psychological nightmare treatments: A systematic review and thematic analysis of discussed hypotheses. ''Sleep Medicine Reviews, 39'', 122–133. https://doi.org/10.1016/j.smrv.2017.08.004

Sandman, N., Valli, K., Kronholm, E., Vartiainen, E., Laatikainen, T., &amp; Paunio, T. (2017). Nightmares as predictors of suicide: An extension study including war veterans. ''Scientific Reports, 7'', 44756. https://doi.org/10.1038/srep44756

Scarpelli, S. (2026). Emotional regulation or just continuity? Rethinking the emotional function of dreaming in the age of computational sleep science. ''Sleep (New York, N.Y.), 49''(5),1. https://doi.org/10.1093/sleep/zsaf375 

Scarpelli, S., Alfonsi, V., Gorgoni, M., &amp; De Gennaro, L. (2022). What about dreams? State of the art and open questions. ''Journal of Sleep Research, 31''(4), e13609. https://doi.org/10.1111/jsr.13609

Schredl, M. (2017). Theorizing about the continuity between waking and dreaming: Comment on Domhoff (2017). ''Dreaming, 27''(4), 351–359. https://doi.org/10.1037/drm0000062

Schwartz, S., Clerget, A., &amp; Perogamvros, L. (2022). Enhancing imagery rehearsal therapy for nightmares with targeted memory reactivation. ''Current Biology, 32''(22), 4808-4816.e4. https://doi.org/10.1016/j.cub.2022.09.032 

Sikka, P., Engelbrektsson, H., Zhang, J., &amp; Gross, J. J. (2022). Negative dream affect is associated with next-day affect level, but not with affect reactivity or affect regulation. ''Frontiers in Behavioral Neuroscience, 16'', 981289. https://doi.org/10.3389/fnbeh.2022.981289

Sikka, P., Feilhauer, D., Valli, K., &amp; Revonsuo, A. (2017). How you measure is what you get: Differences in self- and external ratings of emotional experiences in home dreams. ''The American Journal of Psychology, 130''(3), 367–384. https://doi.org/10.5406/amerjpsyc.130.3.0367

Solms, M. (2000). Dreaming and REM sleep are controlled by different brain mechanisms. ''Behavioral and Brain Sciences, 23''(6), 843–850. https://doi.org/10.1017/S0140525X00003988

Sterpenich, V., Perogamvros, L., Tononi, G., &amp; Schwartz, S. (2020). Fear in dreams and in wakefulness: Evidence for day/night affective homeostasis. ''Human Brain Mapping, 41''(3), 840–850. https://doi.org/10.1002/hbm.24843 

Tousignant, O. H., Glass, D. J., Suvak, M. K., &amp; Fireman, G. D. (2022). Nightmares and nondisturbed dreams impact daily change in negative emotion. ''Dreaming (New York, N.Y.), 32''(3),292-313. Advance online publication. https://doi.org/10.1037/drm0000213 

van der Helm, E., Yao, J., Dutt, S., Rao, V., Saletin, J. M., &amp; Walker, M. P. (2011). REM sleep depotentiates amygdala activity to previous emotional experiences. ''Current Biology, 21''(23), 2029–2032. https://doi.org/10.1016/j.cub.2011.10.052

Walker, M. P., &amp; van der Helm, E. (2009). Overnight therapy? The role of sleep in emotional brain processing. ''Psychological Bulletin, 135''(5), 731–748. https://doi.org/10.1037/a0016570

Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
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      <text bytes="37407" sha1="kw3b9pprhil7iyjojyhmfoyxm85995q" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== The continuity hypothesis ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* Here the dream adds recombination: frightening memories are replayed in new, non-fearful settings (Nielsen &amp; Levin, 2007). [[w:Extinction (psychology)|Extinction]] doesn't erase the fear, it builds a competing safe memory alongside it, which is why the fear returns if that new memory isn't maintained (Nielsen &amp; Levin, 2007).
* On this account nightmares are a breakdown of this process, as the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* Disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (PTSD), which gives the model a clinical test case (Pace-Schott et al., 2015).
* Cathartic dreams, which move from danger to relief, are less common in nightmare disorder than in healthy sleepers, and were the only dream type that tracked falling depression scores across therapy (Perogamvros et al., 2025).
* The main criticism is that the model treats bad dreams as adaptive and nightmares as maladaptive while attributing both to a single process, and it has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007).
* This is the theory tested most directly in ''Using dreams to improve coping'', because sleep interventions manipulate what gets replayed.

===Mood regulation ===

* Here the dream adds desensitisation. Dream emotion is proposed to shift from negative toward positive across the night, so its charge is reduced by morning (Cartwright et al., 1998). This is the headline claim, and the one closest to the popular idea that dreams work through problems, and it is tested in ''Dream emotion and waking life.''
* Dreamer's own ratings of the emotional tone of their day-to-day dreams have been read as inconsistent with a mood regulation function (Barbeau et al., 2022).
* The supporting evidence is real but indirect: people who report more fear in dreams show weaker fear-related brain responses when awake (Sterpenich et al., 2020). That is a correlation across people at a single time point, so it can't establish direction. 
* The theory makes three testable predictions for the morning after a negative dream: mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be better (Sikka et al., 2022).
* Applied to Natalie, it predicts that a dream should be losing intensity over time, and hers is not.

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Barbeau, K., Turpin, C., Lafrenière, A., Campbell, E., &amp; De Koninck, J. (2022). Dreamers' evaluation of the emotional valence of their day-to-day dreams is indicative of some mood regulation function. ''Frontiers in Behavioral Neuroscience, 16'', 947396. https://doi.org/10.3389/fnbeh.2022.947396

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="37674" sha1="56qk7djicqwofw5s5jx17sbpo4vyrzl" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Here the dream adds desensitisation. Dream emotion is proposed to shift from negative toward positive across the night, so its charge is reduced by morning (Cartwright et al., 1998). This is the headline claim, and the one closest to the popular idea that dreams work through problems, and it is tested in ''Dream emotion and waking life.''
* Dreamer's own ratings of the emotional tone of their day-to-day dreams have been read as inconsistent with a mood regulation function (Barbeau et al., 2022).
* The supporting evidence is real but indirect: people who report more fear in dreams show weaker fear-related brain responses when awake (Sterpenich et al., 2020). That is a correlation across people at a single time point, so it can't establish direction. 
* The theory makes three testable predictions for the morning after a negative dream: mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be better (Sikka et al., 2022).
* Applied to Natalie, it predicts that a dream should be losing intensity over time, and hers is not.

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking life ==
'''''Focus question: '''What does dream emotion predict in waking life?''  

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
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__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What does dreaming add beyond REM sleep? 
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict about waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
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===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Malinowski, J. E., &amp; Horton, C. L. (2015). Metaphor and hyperassociativity: The imagination mechanisms behind emotion assimilation in sleep and dreaming. ''Frontiers in Psychology, 6'', 1132. https://doi.org/10.3389/fpsyg.2015.01132

Mendoza Alvarez, M., Balthasar, Y., Verbraecken, J., Claes, L., van Someren, E., van Marle, H. J. F., Vandekerckhove, M., &amp; De Picker, L. (2024). Systematic review: REM sleep, dysphoric dreams and nightmares as transdiagnostic features of psychiatric disorders with emotion dysregulation – Clinical implications. ''Sleep Medicine, 127'', 1–15. https://doi.org/10.1016/j.sleep.2024.12.037

Morgenthaler, T. I., Auerbach, S., Casey, K. R., Kristo, D., Maganti, R., Ramar, K., Zak, R., &amp; Kartje, R. (2018). Position paper for the treatment of nightmare disorder in adults: An American Academy of Sleep Medicine position paper. ''Journal of Clinical Sleep Medicine, 14''(6), 1041–1055. https://doi.org/10.5664/jcsm.7178

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Pace-Schott, E. F., Germain, A., &amp; Milad, M. R. (2015). Sleep and REM sleep disturbance in the pathophysiology of PTSD: The role of extinction memory. ''Biology of Mood &amp; Anxiety Disorders, 5''(1). https://doi.org/10.1186/s13587-015-0018-9 

Palmieri, J., Elce, V., &amp; Schönauer, M. (2025). Nightly dynamics of emotional content in dreams. ''Neuropsychologia, 218''(1), 109248. https://doi.org/10.1016/j.neuropsychologia.2025.109248 

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Schwartz, S., Clerget, A., &amp; Perogamvros, L. (2022). Enhancing imagery rehearsal therapy for nightmares with targeted memory reactivation. ''Current Biology, 32''(22), 4808-4816.e4. https://doi.org/10.1016/j.cub.2022.09.032 

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van der Helm, E., Yao, J., Dutt, S., Rao, V., Saletin, J. M., &amp; Walker, M. P. (2011). REM sleep depotentiates amygdala activity to previous emotional experiences. ''Current Biology, 21''(23), 2029–2032. https://doi.org/10.1016/j.cub.2011.10.052

Walker, M. P., &amp; van der Helm, E. (2009). Overnight therapy? The role of sleep in emotional brain processing. ''Psychological Bulletin, 135''(5), 731–748. https://doi.org/10.1037/a0016570

Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="37994" sha1="dc0753s7s1y28ej37vquodicshtv4xl" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict in waking life?
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict about waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict about waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below predicts a distressing dream after a distressing day, so dream content on its own cannot decide between them. They can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life. 
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict about waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Barbeau, K., Turpin, C., Lafrenière, A., Campbell, E., &amp; De Koninck, J. (2022). Dreamers' evaluation of the emotional valence of their day-to-day dreams is indicative of some mood regulation function. ''Frontiers in Behavioral Neuroscience, 16'', 947396. https://doi.org/10.3389/fnbeh.2022.947396

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict about waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict about waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2022). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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Schredl, M. (2017). Theorizing about the continuity between waking and dreaming: Comment on Domhoff (2017). ''Dreaming, 27''(4), 351–359. https://doi.org/10.1037/drm0000062

Schwartz, S., Clerget, A., &amp; Perogamvros, L. (2022). Enhancing imagery rehearsal therapy for nightmares with targeted memory reactivation. ''Current Biology, 32''(22), 4808-4816.e4. https://doi.org/10.1016/j.cub.2022.09.032 

Sikka, P., Engelbrektsson, H., Zhang, J., &amp; Gross, J. J. (2022). Negative dream affect is associated with next-day affect level, but not with affect reactivity or affect regulation. ''Frontiers in Behavioral Neuroscience, 16'', 981289. https://doi.org/10.3389/fnbeh.2022.981289

Sikka, P., Feilhauer, D., Valli, K., &amp; Revonsuo, A. (2017). How you measure is what you get: Differences in self- and external ratings of emotional experiences in home dreams. ''The American Journal of Psychology, 130''(3), 367–384. https://doi.org/10.5406/amerjpsyc.130.3.0367

Solms, M. (2000). Dreaming and REM sleep are controlled by different brain mechanisms. ''Behavioral and Brain Sciences, 23''(6), 843–850. https://doi.org/10.1017/S0140525X00003988

Sterpenich, V., Perogamvros, L., Tononi, G., &amp; Schwartz, S. (2020). Fear in dreams and in wakefulness: Evidence for day/night affective homeostasis. ''Human Brain Mapping, 41''(3), 840–850. https://doi.org/10.1002/hbm.24843 

Tousignant, O. H., Glass, D. J., Suvak, M. K., &amp; Fireman, G. D. (2022). Nightmares and nondisturbed dreams impact daily change in negative emotion. ''Dreaming (New York, N.Y.), 32''(3),292-313. Advance online publication. https://doi.org/10.1037/drm0000213 

van der Helm, E., Yao, J., Dutt, S., Rao, V., Saletin, J. M., &amp; Walker, M. P. (2011). REM sleep depotentiates amygdala activity to previous emotional experiences. ''Current Biology, 21''(23), 2029–2032. https://doi.org/10.1016/j.cub.2011.10.052

Walker, M. P., &amp; van der Helm, E. (2009). Overnight therapy? The role of sleep in emotional brain processing. ''Psychological Bulletin, 135''(5), 731–748. https://doi.org/10.1037/a0016570

Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="37884" sha1="rddhpkjxu6vl6hjt7koqs79tbqdkp4l" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict about waking coping? 
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict about waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Barbeau, K., Turpin, C., Lafrenière, A., Campbell, E., &amp; De Koninck, J. (2022). Dreamers' evaluation of the emotional valence of their day-to-day dreams is indicative of some mood regulation function. ''Frontiers in Behavioral Neuroscience, 16'', 947396. https://doi.org/10.3389/fnbeh.2022.947396

Blagrove, M., Farmer, L., &amp; Williams, E. (2004). The relationship of nightmare frequency and nightmare distress to well-being. ''Journal of Sleep Research, 13''(2), 129–136. https://doi.org/10.1111/j.1365-2869.2004.00394.x

Cartwright, R., Agargun, M. Y., Kirkby, J., &amp; Friedman, J. K. (2006). Relation of dreams to waking concerns. ''Psychiatry Research, 141''(3), 261–270. https://doi.org/10.1016/j.psychres.2005.05.013

Cartwright, R., Luten, A., Young, M., Mercer, P., &amp; Bears, M. (1998). Role of REM sleep and dream affect in overnight mood regulation: A study of normal volunteers. ''Psychiatry Research, 81''(1), 1–8. https://doi.org/10.1016/S0165-1781(98)00089-4

Casement, M. D., &amp; Swanson, L. M. (2012). A meta-analysis of imagery rehearsal for post-trauma nightmares: Effects on nightmare frequency, sleep quality, and posttraumatic stress. ''Clinical Psychology Review, 32''(6), 566–574. https://doi.org/10.1016/j.cpr.2012.06.002

Cook, J. M., Harb, G. C., Gehrman, P. R., Cary, M. S., Gamble, G. M., Forbes, D., &amp; Ross, R. J. (2010). Imagery rehearsal for posttraumatic nightmares: A randomized controlled trial. ''Journal of Traumatic Stress, 23''(5), 553–563. https://doi.org/10.1002/jts.20569

Desjardins, S., &amp; Zadra, A. (2006). Is the threat simulation theory threatened by recurrent dreams? ''Consciousness and Cognition, 15''(2), 470–474. https://doi.org/10.1016/j.concog.2006.04.005

Domhoff, G. W. (2017). The invasion of the concept snatchers: The origins, distortions, and future of the continuity hypothesis. ''Dreaming, 27''(1), 14–39. https://doi.org/10.1037/drm0000047

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Folkman, S., &amp; Moskowitz, J. T. (2004). Coping: Pitfalls and promise. ''Annual Review of Psychology, 55''(1), 745–774. https://doi.org/10.1146/annurev.psych.55.090902.141456

Gauchat, A., Zadra, A., El-Hourani, M., Parent, S., Tremblay, R. E., &amp; Séguin, J. R. (2020). Disturbing dreams and psychosocial maladjustment in children: A prospective study of the moderating role of early negative emotionality. ''Frontiers in Neurology, 11'', 762. https://doi.org/10.3389/fneur.2020.00762

Gieselmann, A., Ait Aoudia, M., Carr, M., Germain, A., Gorzka, R., Holzinger, B., Kleim, B., Krakow, B., Kunze, A. E., Lancee, J., Nadorff, M. R., Nielsen, T., Riemann, D., Sandahl, H., Schlarb, A. A., Schmid, C., Schredl, M., Spoormaker, V. I., Steil, R., … Pietrowsky, R. (2019). Aetiology and treatment of nightmare disorder: State of the art and future perspectives. ''Journal of Sleep Research, 28''(4), e12820. https://doi.org/10.1111/jsr.12820

Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry, 26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Haar Horowitz, A., Cunningham, T. J., Maes, P., &amp; Stickgold, R. (2020). Dormio: A targeted dream incubation device. ''Consciousness and Cognition, 83'', 102938. https://doi.org/10.1016/j.concog.2020.102938

Harb, G. C., Cook, J. M., Phelps, A. J., Gehrman, P. R., Forbes, D., Localio, R., Harpaz-Rotem, I., Gur, R. C., &amp; Ross, R. J. (2019). Randomized controlled trial of imagery rehearsal for posttraumatic nightmares in combat veterans. ''Journal of Clinical Sleep Medicine, 15''(5), 757–767. https://doi.org/10.5664/jcsm.7770

Kempe, S., Köpp, W., &amp; Wittmann, L. (2024). Personality functioning improvement during psychotherapy is associated with an enhanced capacity for affect regulation in dreams: A preliminary study. ''Brain Sciences, 14''(5), 489. https://doi.org/10.3390/brainsci14050489

Lara-Carrasco, J., Nielsen, T. A., Solomonova, E., Levrier, K., &amp; Popova, A. (2009). Overnight emotional adaptation to negative stimuli is altered by REM sleep deprivation and is correlated with intervening dream emotions. ''Journal of Sleep Research, 18''(2), 178–187. https://doi.org/10.1111/j.1365-2869.2008.00709.x

Lipinska, G., Austin, H., Moonsamy, J. R., Henry, M., Lewis, R., Baldwin, D. S., Thomas, K. G. F., &amp; Stuart, B. (2022). Preferential consolidation of emotional reactivity during sleep: A systematic review and meta-analysis. ''Frontiers in Behavioral Neuroscience, 16'', 976047. https://doi.org/10.3389/fnbeh.2022.976047

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Malinowski, J. E., &amp; Horton, C. L. (2015). Metaphor and hyperassociativity: The imagination mechanisms behind emotion assimilation in sleep and dreaming. ''Frontiers in Psychology, 6'', 1132. https://doi.org/10.3389/fpsyg.2015.01132

Mendoza Alvarez, M., Balthasar, Y., Verbraecken, J., Claes, L., van Someren, E., van Marle, H. J. F., Vandekerckhove, M., &amp; De Picker, L. (2024). Systematic review: REM sleep, dysphoric dreams and nightmares as transdiagnostic features of psychiatric disorders with emotion dysregulation – Clinical implications. ''Sleep Medicine, 127'', 1–15. https://doi.org/10.1016/j.sleep.2024.12.037

Morgenthaler, T. I., Auerbach, S., Casey, K. R., Kristo, D., Maganti, R., Ramar, K., Zak, R., &amp; Kartje, R. (2018). Position paper for the treatment of nightmare disorder in adults: An American Academy of Sleep Medicine position paper. ''Journal of Clinical Sleep Medicine, 14''(6), 1041–1055. https://doi.org/10.5664/jcsm.7178

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Pace-Schott, E. F., Germain, A., &amp; Milad, M. R. (2015). Sleep and REM sleep disturbance in the pathophysiology of PTSD: The role of extinction memory. ''Biology of Mood &amp; Anxiety Disorders, 5''(1). https://doi.org/10.1186/s13587-015-0018-9 

Palmieri, J., Elce, V., &amp; Schönauer, M. (2025). Nightly dynamics of emotional content in dreams. ''Neuropsychologia, 218''(1), 109248. https://doi.org/10.1016/j.neuropsychologia.2025.109248 

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Sterpenich, V., Perogamvros, L., Tononi, G., &amp; Schwartz, S. (2020). Fear in dreams and in wakefulness: Evidence for day/night affective homeostasis. ''Human Brain Mapping, 41''(3), 840–850. https://doi.org/10.1002/hbm.24843 

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van der Helm, E., Yao, J., Dutt, S., Rao, V., Saletin, J. M., &amp; Walker, M. P. (2011). REM sleep depotentiates amygdala activity to previous emotional experiences. ''Current Biology, 21''(23), 2029–2032. https://doi.org/10.1016/j.cub.2011.10.052

Walker, M. P., &amp; van der Helm, E. (2009). Overnight therapy? The role of sleep in emotional brain processing. ''Psychological Bulletin, 135''(5), 731–748. https://doi.org/10.1037/a0016570

Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
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__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict about waking coping? 
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
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===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
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==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Kempe, S., Köpp, W., &amp; Wittmann, L. (2024). Personality functioning improvement during psychotherapy is associated with an enhanced capacity for affect regulation in dreams: A preliminary study. ''Brain Sciences, 14''(5), 489. https://doi.org/10.3390/brainsci14050489

Lara-Carrasco, J., Nielsen, T. A., Solomonova, E., Levrier, K., &amp; Popova, A. (2009). Overnight emotional adaptation to negative stimuli is altered by REM sleep deprivation and is correlated with intervening dream emotions. ''Journal of Sleep Research, 18''(2), 178–187. https://doi.org/10.1111/j.1365-2869.2008.00709.x

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Mendoza Alvarez, M., Balthasar, Y., Verbraecken, J., Claes, L., van Someren, E., van Marle, H. J. F., Vandekerckhove, M., &amp; De Picker, L. (2024). Systematic review: REM sleep, dysphoric dreams and nightmares as transdiagnostic features of psychiatric disorders with emotion dysregulation – Clinical implications. ''Sleep Medicine, 127'', 1–15. https://doi.org/10.1016/j.sleep.2024.12.037

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Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* This evidence here points one way: dream emotion tracks distress rather than relieving it, and the mood regulation predictions largely fail.
* Dreaming only counts as coping if dream emotion predicts something measurable while the person is awake (Folkman &amp; Moskowitz, 2004). One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count as regulation, so this doesn't rule dreaming out (Gross, 2015).
* Almost all the evidence below is correlational, which is why the experimental interventions in the next section carry more weight (Peltz et al., 2026) Evidence is examined at three points: the next morning, a major life stressor, and the point at which dreaming becomes a clinical problem.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Dream emotion and next-day mood ===

*Across five days in 40 adults, more negative dream emotion predicted a more negative morning, with no effect on reactivity to negative images or on the ability to down-regulate them (Sikka et al., 2022).
*All three mood predictions therefore failed, and the one effect that emerged ran opposite to the theory. 
*A much larger study found the same: across 4,715 days from 536 adults, frightening dreams were followed by more negative mood the next morning (Baber et al., 2026).
*The same study found the reverse between people, with those who had more frightening dreams overall regulating emotion better when awake (Baber et al., 2026). Dream fear may therefore mark who copes well rather than support their coping. 
*Nights with no remembered dream showed no change in negative emotion at all, which is one of the few findings  suggesting the dream itself adds something beyond REM sleep (Tousignant et al., 2022).
*Results also depend on who rates them: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). Some disagreement between studies above may be disagreement about measurement. 

===Dreaming and major life stress ===

* The diary studies above followed people with little to regulate, so this sub-section asks what happens when there is real distress (Cartwright et al., 1998).
* Over five months of REM awakenings in adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams lacked emotion and were not linked to other memories (Cartwright et al., 2006).
* Those who recovered had emotional dreams about the same subject, so it is the emotion in the dream rather than the topic that tracks recovery (Cartwright et al., 2006).
* This is the clearest evidence in the chapter that dreaming aids coping with a real stressor, and also the most fragile: one research program, with no replication using modern methods in the twenty five years since (Cartwright et al., 2006).
* Applied to Natalie, it predicts that feeling the loss in the dream is a better sign than dreaming about it without emotion.

===Nightmares and distress ===

* The two sections above asked whether dreaming helps, this subsection asks what happens when it clearly does not.
* Nightmares occur weekly in roughly 4 to 10% of people, and about 4% of adults meet the criteria for nightmare disorder, so this is a common problem rather than a rare one (Nielsen &amp; Levin, 2007; Morgenthaler et al., 2018). 
* Two things produce them: emotional pressure supplies the material, and a tendency to react strongly makes dreams distressing (Nielsen &amp; Levin, 2007).
* The second matters more because distress predicts wellbeing better than frequency does (Blagrove et al., 2004). This is why treatment targets distress, and why Natalie's dream is a problem even though she sleeps enough. 
* Nightmares predicted [[w:Suicidal ideation|suicidal ideation]] in 583 students independently of insomnia, depression, anxiety and PTSD, and predicted death by suicide in 71,068 Finnish adults (Nadorff et al., 2011; Sandman et al., 2017). 
* They appear across disorders involving emotion dysregulation, and in children they prospectively predict later psychosocial maladjustment (Mendoza Alvarez et al., 2024; Gauchat et al., 2020).
* Nightmare disorder is now treated as a condition in its own right rather than only a symptom of something else, which is where section 4 begins (Gieselmann et al., 2019).

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
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===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
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==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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Pesonen, A.-K., Koskinen, M.-K., Vuorenhela, N., Halonen, R., Mäkituuri, S., Selin, M., Luokkala, S., Suutari, A., &amp; Hovatta, I. (2024). The effect of REM-sleep disruption on affective processing: A systematic review of human and animal experimental studies. ''Neuroscience &amp; Biobehavioral Reviews, 162'', 105714. https://doi.org/10.1016/j.neubiorev.2024.105714

Revonsuo, A. (2000). The reinterpretation of dreams: An evolutionary hypothesis of the function of dreaming. ''Behavioral and Brain Sciences, 23''(6), 877–901. https://doi.org/10.1017/S0140525X00004015

Rousseau, A., &amp; Belleville, G. (2018). The mechanisms of action underlying the efficacy of psychological nightmare treatments: A systematic review and thematic analysis of discussed hypotheses. ''Sleep Medicine Reviews, 39'', 122–133. https://doi.org/10.1016/j.smrv.2017.08.004

Sandman, N., Valli, K., Kronholm, E., Vartiainen, E., Laatikainen, T., &amp; Paunio, T. (2017). Nightmares as predictors of suicide: An extension study including war veterans. ''Scientific Reports, 7'', 44756. https://doi.org/10.1038/srep44756

Scarpelli, S. (2026). Emotional regulation or just continuity? Rethinking the emotional function of dreaming in the age of computational sleep science. ''Sleep (New York, N.Y.), 49''(5),1. https://doi.org/10.1093/sleep/zsaf375 

Scarpelli, S., Alfonsi, V., Gorgoni, M., &amp; De Gennaro, L. (2022). What about dreams? State of the art and open questions. ''Journal of Sleep Research, 31''(4), e13609. https://doi.org/10.1111/jsr.13609

Schredl, M. (2017). Theorizing about the continuity between waking and dreaming: Comment on Domhoff (2017). ''Dreaming, 27''(4), 351–359. https://doi.org/10.1037/drm0000062

Schwartz, S., Clerget, A., &amp; Perogamvros, L. (2022). Enhancing imagery rehearsal therapy for nightmares with targeted memory reactivation. ''Current Biology, 32''(22), 4808-4816.e4. https://doi.org/10.1016/j.cub.2022.09.032 

Sikka, P., Engelbrektsson, H., Zhang, J., &amp; Gross, J. J. (2022). Negative dream affect is associated with next-day affect level, but not with affect reactivity or affect regulation. ''Frontiers in Behavioral Neuroscience, 16'', 981289. https://doi.org/10.3389/fnbeh.2022.981289

Sikka, P., Feilhauer, D., Valli, K., &amp; Revonsuo, A. (2017). How you measure is what you get: Differences in self- and external ratings of emotional experiences in home dreams. ''The American Journal of Psychology, 130''(3), 367–384. https://doi.org/10.5406/amerjpsyc.130.3.0367

Solms, M. (2000). Dreaming and REM sleep are controlled by different brain mechanisms. ''Behavioral and Brain Sciences, 23''(6), 843–850. https://doi.org/10.1017/S0140525X00003988

Sterpenich, V., Perogamvros, L., Tononi, G., &amp; Schwartz, S. (2020). Fear in dreams and in wakefulness: Evidence for day/night affective homeostasis. ''Human Brain Mapping, 41''(3), 840–850. https://doi.org/10.1002/hbm.24843 

Tousignant, O. H., Glass, D. J., Suvak, M. K., &amp; Fireman, G. D. (2022). Nightmares and nondisturbed dreams impact daily change in negative emotion. ''Dreaming (New York, N.Y.), 32''(3),292-313. Advance online publication. https://doi.org/10.1037/drm0000213 

van der Helm, E., Yao, J., Dutt, S., Rao, V., Saletin, J. M., &amp; Walker, M. P. (2011). REM sleep depotentiates amygdala activity to previous emotional experiences. ''Current Biology, 21''(23), 2029–2032. https://doi.org/10.1016/j.cub.2011.10.052

Walker, M. P., &amp; van der Helm, E. (2009). Overnight therapy? The role of sleep in emotional brain processing. ''Psychological Bulletin, 135''(5), 731–748. https://doi.org/10.1037/a0016570

Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="37669" sha1="5wxj523pzbcz9mpp3oe5bi71g3hdrzb" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does it. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022). 
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
Baber, G. R., Hamilton, N. A., Girard, J. M., Sikka, P., Watts, A., Jun, D., Gratton, M. K. P., Cohen, J. M., Quesada, A. K., Nichols, E. C., &amp; Cunningham, T. J. (2026). Testing affect regulation theories of dreaming. ''Sleep, 49''(5). https://doi.org/10.1093/sleep/zsag046 

Barbeau, K., Turpin, C., Lafrenière, A., Campbell, E., &amp; De Koninck, J. (2022). Dreamers' evaluation of the emotional valence of their day-to-day dreams is indicative of some mood regulation function. ''Frontiers in Behavioral Neuroscience, 16'', 947396. https://doi.org/10.3389/fnbeh.2022.947396

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Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does it. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022). 
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

=== What coping research is still missing ===

* All three tests above use mood, reactivity, or recovery but none measure what a person actually does to manage stress, which is what coping means (Folkman &amp; Moskowitz, 2004). 
* The one study to measure both found that people who used positive reappraisal while awake had more problem-solving in their dreams, but that those who dreamt about the stressor reported less problem solving awake (Delorme et al., 2002). It used 22 dreamers and a mild stressor, and it is close to the whole literature connecting dreams to waking coping behaviour. 
* None of this can settle whether the dream causes anything, because no study here could change a dream, the next section does. 
&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* This section concludes that dreams can be changed deliberately and that doing so helps, though it is unclear whether the change to the dream is what helps.
* That combination is the chapters central point: a dream can be a useful tool without being an automatic therapist. 
* These are the only randomised experiments in the chapter, so they carry the most casual weight (Gieselmann et al., 2019).

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does it. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022). 
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

=== What coping research is still missing ===

* All three tests above use mood, reactivity, or recovery but none measure what a person actually does to manage stress, which is what coping means (Folkman &amp; Moskowitz, 2004). 
* The one study to measure both found that people who used positive reappraisal while awake had more problem-solving in their dreams, but that those who dreamt about the stressor reported less problem solving awake (Delorme et al., 2002). It used 22 dreamers and a mild stressor, and it is close to the whole literature connecting dreams to waking coping behaviour. 
* None of this can settle whether the dream causes anything, because no study here could change a dream, the next section does. 
&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* The previous section found that dream emotion tarcks how well someone is coping without clear evidence that it drives it, which may be a real limit or only a limit of the method. 
* This section reverse the direction of the measurement, instead of recording a dream and looking forward to the morning, it changes something while the person is awake and then looks back at the dream. That is the only order that can show cause. 

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Rewriting the dream: imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) involves writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018). 
* The logic follows the fear extinction section: rehearsing a new ending does while awake what fear extinction says a healthy dream does at night.
* IRT is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and a network meta-analysis ranks it among the most effective options (Y. Zhang et al., 2022).
* Meta-analysis finds large improvements held at 6 to 12 months, but these are the before and after effects rather than controlled ones (Casement &amp; Swanson, 2012).
* Two trials found no advantage for the rewriting itself over other sleep treatments (Cook et al., 2010; Harb et al., 2019), and why IRT works is still disputed (Rousseau &amp; Belleville, 2018).
* If it works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

===Intervening during sleep itself ===

*The trials above leave one question: does anything happen during sleep that waking rehearsal cannot do alone?
* Targeted memory reactivation tests this by pairing the new ending with a sound while awake, then replaying the sound quietly during REM sleep (Schwartz et al., 2022).
* Adding the sound beat IRT alone, making this the strongest casual evidence in the chapter and the closest test of fear extinction (Schwartz et al., 2022). Though it is one small trial and needs replication. 
* A second approach plants dream content rather than cueing it: targeted dream incubation repeats a chosen theme at sleep onset, and 67% of prompted awakenings produced dream reports containing it, against 3% without the prompt (Haar Horowitz et al., 2020). 
* The point is methodological: a causal claim about dreaming requires manipulating dream content, which nothing in sections 1 and 3 does. Even a successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work (Haar Horowitz et al., 2020).
* Working the other way, better emotion regulation within dreams accompanied improvement during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="38075" sha1="nhn0p10q2bicb6c05lcgxj29dskkowp" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

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'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

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==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does it. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022). 
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

=== What coping research is still missing ===

* All three tests above use mood, reactivity, or recovery but none measure what a person actually does to manage stress, which is what coping means (Folkman &amp; Moskowitz, 2004). 
* The one study to measure both found that people who used positive reappraisal while awake had more problem-solving in their dreams, but that those who dreamt about the stressor reported less problem solving awake (Delorme et al., 2002). It used 22 dreamers and a mild stressor, and it is close to the whole literature connecting dreams to waking coping behaviour. 
* None of this can settle whether the dream causes anything, because no study here could change a dream, the next section does. 
&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* The previous section found that dream emotion tarcks how well someone is coping without clear evidence that it drives it, which may be a real limit or only a limit of the method. 
* This section reverse the direction of the measurement, instead of recording a dream and looking forward to the morning, it changes something while the person is awake and then looks back at the dream. That is the only order that can show cause. 

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
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===Imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) means writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018).
* The logic follows fear extinction: rehearsing a new ending does while awake what the theory says a healthy dream does at night.
* It is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and across 29 randomised trials it was one of the only two treatments to beat control conditions (Y. Zhang et al., 2022).
* Why IRT works remains disputed: a systematic review of the proposed mechanisms found none established (Rousseau &amp; Belleville, 2018), and 30% of patients did not respond at all (Schwartz et al., 2022). 
* Implication: If IRT works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

=== Manipulating dream content: the missing method ===

* A causal claim about dreaming requires manipulating dream content, which nothing in the previous sections does. Targeted dream incubation repeats a chosen theme at sleep onset: 67% of prompted awakenings produced dream reports containing the theme, against 3% of unprompted awakenings (Haar Horowitz et al., 2020). Dream content is therefore controllable. 
* But this is a methods demonstration rather than a treatment, even successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work being done (Haar Horowitz et al., 2020). 
* Working the other way, an improved capacity for affect regulation within dreams accompanied improvement in personality functioning during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).
* What would settle the question is an experiment that manipulates dream emotion under real distress and measures waking coping rather than nightmare frequency. Nothing in this literature does that yet. 

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

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'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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      <text bytes="38453" sha1="g5lqcwe5jc28whksy3h0wbo952pvql4" xml:space="preserve">{{title|Dreams and emotional problem-solving -&lt;br&gt;How do REM dreams contribute to emotional processing and adaptive coping?}}
__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
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* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does not. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022).
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

=== What coping research is still missing ===

* All three tests above use mood, reactivity, or recovery but none measure what a person actually does to manage stress, which is what coping means (Folkman &amp; Moskowitz, 2004). 
* The one study to measure both found that people who used positive reappraisal while awake had more problem-solving in their dreams, but that those who dreamt about the stressor reported less problem solving awake (Delorme et al., 2002). It used 22 dreamers and a mild stressor, and it is close to the whole literature connecting dreams to waking coping behaviour. 
* None of this can settle whether the dream causes anything, because no study here could change a dream, the next section does. 
&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* The previous section found that dream emotion tarcks how well someone is coping without clear evidence that it drives it, which may be a real limit or only a limit of the method. 
* This section reverse the direction of the measurement, instead of recording a dream and looking forward to the morning, it changes something while the person is awake and then looks back at the dream. That is the only order that can show cause. 

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) means writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018).
* The logic follows fear extinction: rehearsing a new ending does while awake what the theory says a healthy dream does at night.
* It is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and across 29 randomised trials it was one of the only two treatments to beat control conditions (Y. Zhang et al., 2022).
* The large effects are uncontrolled: improvements in nightmares, sleep and PTSD held at 6 and 12 months across 13 studies but shrank to a small effect in the one trial that compared it against another active treatment (Casement &amp; Swanson, 2012), and two later trials found no advantage over other sleep treatments in veterans with PTSD (Cook et al., 2010; Harb et al., 2019). 
* Why IRT works remains disputed: a systematic review of the proposed mechanisms found none established (Rousseau &amp; Belleville, 2018), and 30% of patients did not respond at all (Schwartz et al., 2022). 
* Implication: If IRT works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

=== Manipulating dream content: the missing method ===

* A causal claim about dreaming requires manipulating dream content, which nothing in the previous sections does. Targeted dream incubation repeats a chosen theme at sleep onset: 67% of prompted awakenings produced dream reports containing the theme, against 3% of unprompted awakenings (Haar Horowitz et al., 2020). Dream content is therefore controllable. 
* But this is a methods demonstration rather than a treatment, even successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work being done (Haar Horowitz et al., 2020). 
* Working the other way, an improved capacity for affect regulation within dreams accompanied improvement in personality functioning during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).
* What would settle the question is an experiment that manipulates dream emotion under real distress and measures waking coping rather than nightmare frequency. Nothing in this literature does that yet. 

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep changes emotional memories: it makes them more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019). Whether the dream contributes to that change is a separate question, and only two studies have tested it directly (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for, they cannot be seperated by what dreams contain, only by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it, because a causal claim about dreaming requires dreams to be manipulated (Haar Horowitz et al., 2020; Schwartz et al., 2022).
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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[[Category:Motivation and emotion/Book/Dreams]]
[[Category:Motivation and emotion/Book/Emotion]]</text>
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__TOC__
==Overview==
{{RoundBoxTop|theme=11}}
'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does not. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022).
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

=== What coping research is still missing ===

* All three tests above use mood, reactivity, or recovery but none measure what a person actually does to manage stress, which is what coping means (Folkman &amp; Moskowitz, 2004). 
* The one study to measure both found that people who used positive reappraisal while awake had more problem-solving in their dreams, but that those who dreamt about the stressor reported less problem solving awake (Delorme et al., 2002). It used 22 dreamers and a mild stressor, and it is close to the whole literature connecting dreams to waking coping behaviour. 
* None of this can settle whether the dream causes anything, because no study here could change a dream, the next section does. 
&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* The previous section found that dream emotion tarcks how well someone is coping without clear evidence that it drives it, which may be a real limit or only a limit of the method. 
* This section reverse the direction of the measurement, instead of recording a dream and looking forward to the morning, it changes something while the person is awake and then looks back at the dream. That is the only order that can show cause. 

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'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) means writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018).
* The logic follows fear extinction: rehearsing a new ending does while awake what the theory says a healthy dream does at night.
* It is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and across 29 randomised trials it was one of the only two treatments to beat control conditions (Y. Zhang et al., 2022).
* The large effects are uncontrolled: improvements in nightmares, sleep and PTSD held at 6 and 12 months across 13 studies but shrank to a small effect in the one trial that compared it against another active treatment (Casement &amp; Swanson, 2012), and two later trials found no advantage over other sleep treatments in veterans with PTSD (Cook et al., 2010; Harb et al., 2019). 
* Why IRT works remains disputed: a systematic review of the proposed mechanisms found none established (Rousseau &amp; Belleville, 2018), and 30% of patients did not respond at all (Schwartz et al., 2022). 
* Implication: If IRT works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

=== Manipulating dream content: the missing method ===

* A causal claim about dreaming requires manipulating dream content, which nothing in the previous sections does. Targeted dream incubation repeats a chosen theme at sleep onset: 67% of prompted awakenings produced dream reports containing the theme, against 3% of unprompted awakenings (Haar Horowitz et al., 2020). Dream content is therefore controllable. 
* But this is a methods demonstration rather than a treatment, even successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work being done (Haar Horowitz et al., 2020). 
* Working the other way, an improved capacity for affect regulation within dreams accompanied improvement in personality functioning during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).
* What would settle the question is an experiment that manipulates dream emotion under real distress and measures waking coping rather than nightmare frequency. Nothing in this literature does that yet. 

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep and dreaming are not the same thing. Sleep makes emotional memories more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019), and dreaming continues when REM sleep is supressed (Solms, 2000). Only two studies have tested whether the dream itself adds anything (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for and they can be separated by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one, and predicts nothing about reactivity or about the ability to regulate emotion (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Coping itself has almost never been measured, every finding above uses mood, reactivity or clinical recovery as a stand in for what a person actually does to manage stress (Folkman &amp; Moskowitz, 2004; Delorme et al., 2002). 
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it (Haar Horowitz et al., 2020; Schwartz et al., 2022), and measuring coping rather than nightmare frequency. 
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=
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Barbeau, K., Turpin, C., Lafrenière, A., Campbell, E., &amp; De Koninck, J. (2022). Dreamers' evaluation of the emotional valence of their day-to-day dreams is indicative of some mood regulation function. ''Frontiers in Behavioral Neuroscience, 16'', 947396. https://doi.org/10.3389/fnbeh.2022.947396

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Cartwright, R., Luten, A., Young, M., Mercer, P., &amp; Bears, M. (1998). Role of REM sleep and dream affect in overnight mood regulation: A study of normal volunteers. ''Psychiatry Research, 81''(1), 1–8. https://doi.org/10.1016/S0165-1781(98)00089-4

Casement, M. D., &amp; Swanson, L. M. (2012). A meta-analysis of imagery rehearsal for post-trauma nightmares: Effects on nightmare frequency, sleep quality, and posttraumatic stress. ''Clinical Psychology Review, 32''(6), 566–574. https://doi.org/10.1016/j.cpr.2012.06.002

Cook, J. M., Harb, G. C., Gehrman, P. R., Cary, M. S., Gamble, G. M., Forbes, D., &amp; Ross, R. J. (2010). Imagery rehearsal for posttraumatic nightmares: A randomized controlled trial. ''Journal of Traumatic Stress, 23''(5), 553–563. https://doi.org/10.1002/jts.20569

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Harb, G. C., Cook, J. M., Phelps, A. J., Gehrman, P. R., Forbes, D., Localio, R., Harpaz-Rotem, I., Gur, R. C., &amp; Ross, R. J. (2019). Randomized controlled trial of imagery rehearsal for posttraumatic nightmares in combat veterans. ''Journal of Clinical Sleep Medicine, 15''(5), 757–767. https://doi.org/10.5664/jcsm.7770

Kempe, S., Köpp, W., &amp; Wittmann, L. (2024). Personality functioning improvement during psychotherapy is associated with an enhanced capacity for affect regulation in dreams: A preliminary study. ''Brain Sciences, 14''(5), 489. https://doi.org/10.3390/brainsci14050489

Lara-Carrasco, J., Nielsen, T. A., Solomonova, E., Levrier, K., &amp; Popova, A. (2009). Overnight emotional adaptation to negative stimuli is altered by REM sleep deprivation and is correlated with intervening dream emotions. ''Journal of Sleep Research, 18''(2), 178–187. https://doi.org/10.1111/j.1365-2869.2008.00709.x

Lipinska, G., Austin, H., Moonsamy, J. R., Henry, M., Lewis, R., Baldwin, D. S., Thomas, K. G. F., &amp; Stuart, B. (2022). Preferential consolidation of emotional reactivity during sleep: A systematic review and meta-analysis. ''Frontiers in Behavioral Neuroscience, 16'', 976047. https://doi.org/10.3389/fnbeh.2022.976047

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Mendoza Alvarez, M., Balthasar, Y., Verbraecken, J., Claes, L., van Someren, E., van Marle, H. J. F., Vandekerckhove, M., &amp; De Picker, L. (2024). Systematic review: REM sleep, dysphoric dreams and nightmares as transdiagnostic features of psychiatric disorders with emotion dysregulation – Clinical implications. ''Sleep Medicine, 127'', 1–15. https://doi.org/10.1016/j.sleep.2024.12.037

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}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

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__TOC__
==Overview==
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'''Case study: part 1 ''' [[File:Dreaming Woman.jpg|right|thumb|300px|'''Figure 1'''. A recurring dream after a loss keeps its emotional charge overnight, which is exactly what emotional processing during sleep is supposed to reduce.]]

Natalie hasn't slept well in weeks. It's not like she's waking up screaming from a nightmare or anything that dramatic. But the same dream keeps coming back: she has been left behind somewhere, calling out to someone who doesn't turn around. Her friends say it's just stress from the breakup. By lunchtime she can barely remember the details, but the feeling hangs around all day. She's isn't short on sleep, so why does she keep waking up feeling like she's already lost something? Is the dream working through her breakup, or just replaying it? 
{{RoundBoxBottom}}
* A [[w:Dream|dream]] is a conscious experience during sleep that can be recalled on waking, and dreams reported after [[w:Rapid eye movement sleep|rapid eye movement (REM) sleep]] tend to be the most vivid and the most emotional (Scarpelli et al., 2022).
* Two processes are involved here, and it helps to keep them apart. Emotional processing happens overnight, when an emotional memory loses some of its intensity. [[Motivation and emotion/Book/2020/Coping and emotion|Coping]] happens while a person is awake, through the strategies a person uses to manage stress (Folkman &amp; Moskowitz, 2004). The subtitle asks whether the first supports the second.
* Sleep is widely described as overnight therapy: it holds onto the emotional memory but strips some of its sting by morning (Walker &amp; van der Helm, 2009).
* Very few researchers dispute that sleep does something, what they dispute is the role of the dream. Either the dream is where the emotional work happens, or it is a window into work that sleep would be doing anyway (Scarpelli, 2026).
* Why does settling this matter? Because recurring distressing dreams like Natalie's (see Figure 1) are common, they appear across psychiatric disorders involving [[w:Emotional dysregulation|emotion dysregulation]], and they can be treated (Casement &amp; Swanson, 2012; Mendoza Alvarez et al., 2024).

{{RoundBoxTop|theme=15}}
'''Focus questions'''

* Does emotional processing during sleep depend on dreaming?
* What might a dream be doing with emotion?
* What does dream emotion predict for waking coping? 
* How can dreams be targeted to improve coping?  

{{RoundBoxBottom}}

==Dreaming and emotional processing during sleep ==

'''''Focus question:''' Does emotional processing during sleep depend on dreaming?''  
* Natalie's dream keeps coming back with its emotional charge intact, which is what REM sleep is supposed to prevent. Explaining why means separating two things that are easily confused: what REM sleep does to an emotional memory, and what the dream does. Most evidence addresses REM sleep, so this section covers that briefly and then asks what is known about dreaming itself. 

===What sleep does ===

* The leading model is Walker and van der Helm's (2009) &quot;sleep to forget, sleep to remember&quot;. During REM sleep, the brain replays [[Motivation and emotion/Book/2013/Memory and emotion|emotional memories]] while [[w:Norepinephrine|noradrenaline]] (the chemical behind the body's stress response) is switched off, so the memory is kept but the alarm attached to it fades. Note what this predicts: physical arousal should fall while the memory stays intact, so the proper test is reactivity rather than whether the memory feels less bad. 
* Supporting evidence: 34 adults rated the same emotional images 12 hours apart. Reactivity fell across a night and rose across a day, and the largest drop was in those whose noradrenaline was lowest during REM (van der Helm et al., 2011).
* Two stronger tests disagree and they use different methods, which makes it harder to miss. A meta-analysis of 24 studies found negative images rated as more arousing after sleep (Lipinska et al., 2022), and an experiment found participants with the least REM sleep showed the greatest overnight drop in arousal (Lara-Carrasco et al., 2009).
* So sleep preserves emotional memories more reliably than it soothes them (Lipinska et al., 2019), and the model has not yet been properly tested, because removing REM disturbs every over sleep stage as well (Pesonen et al., 2024). 

===What dreaming adds ===

* Dreaming is not simply the conscious experience of REM sleep. Across 111 published cases, forebrain damage stopped dreaming while REM sleep continued normally (Solms, 2000), and dreaming survives drugs that suppress REM (Scarpelli et al., 2026). 
* Because dreaming occurs outside REM sleep, evidence about REM sleep cannot be read as evidence about REM dreams, so dreaming has to be measured directly (Solms, 2000). 
* Measuring dreams is difficult: across 552 home dreams, dreamers rated their own as mostly positive while judges rated the same dreams as mostly negative (Sikka et al., 2017). So studies using different rating sources may not be comparable. 
* Very few studies measure both REM and dreaming in the same people. In one, participants whose dreams were less negative showed the greatest overnight improvement in their ratings of emotional images (Lara-Carrasco et al., 2009). In another, the overnight drop in reactivity appeared only in participants who recalled a dream (Zhang et al., 2024). 
* No study can assign anyone to dream, so the direction is unresolved. Someone already coping well might both dream less negatively and adapt better overnight, with the dream contributing nothing, and mood and REM sleep appear to influence each other in both directions (Peltz et al., 2026). 
* Recall creates a further problem. Whether a dream is remembered depends on the state the sleeper wakes from (Scarpelli et al., 2022), and non-recallers may have dreamed and forgotten, which makes the claim difficult to disprove. 
* Whether dreams regulate emotion, mirror waking concerns, or do both remains an open question (Scarpelli, 2026). Answering it requires manipulating dream content rather than recording it (Haar Horowitz et al., 2020), which is discussed further in the final section. 

==Theories of emotional dream function==
'''''Focus question: '''What might a dream be doing with emotion?''  

*Each theory below can be separated in two ways: the first is the general shape of the dream, as the theories disagree about what a useful dream should look like on the inside, and the second is the next morning, because a dream that did emotional work overnight should leave a trace in waking life.
*Three of the four theories below claim the dream does something while one claims it does not, and that is the position the other three have to beat. Two of the three also predict the same next morning, so four theories make three testable predictions (see Table 1). This matters later, because one result about next-day mood can point towards two theories at once. 
*[[w:Nightmare|Nightmares]] are clearest way to test each theory, because the same dream counts as either the system working for one and failing for another.

{| class=&quot;wikitable&quot;
|+ '''Table 1'''&lt;br /&gt; ''Competing theoretical accounts of the emotional function of dreaming and their predictions''
|-
! Theory !! What the dream does to the feeling !! Next-day mood after a distressing dream !! What a nightmare is 
|-
| '''Continuity''' || Nothing, it expresses the feeling || Unchanged, the dream mirrors mood without moving it || A marker of distress, and nothing more 
|-
| '''Threat simulation''' || Practises with it || No better or possibly worse, the payoff is preparedness || The system working as designed
|-
| '''Fear extinction''' || Rebuilds around it || Better if extinction succeeds, unchanged or worse if it fails, which makes the prediction hard to falsify || The replay failing to build a safe version 
|-
| '''Mood regulation''' || Drains it || Better: less negative mood, weaker reactivity, easier regulation || Regulation failing to complete 
|-
|}

=== Continuity ===

* Dreams express a person's ongoing concerns and leave them exactly as they found them, so the dream is meaningful but it is simply not doing anything (Domhoff, 2017). 
* This is the baseline the other theories have to beat, because showing that a dream is meaningful does not automatically prove that it has a function (Domhoff, 2019).
* Continuity is often misread as the idea that dreams replay yesterday. Domhoff (2017) argues the opposite: across three studies, independent judges could not match dream reports to the events or concerns participants had reported before sleep, so the dream tracks the person's current preoccupations rather than their day. Schredl (2017) replies that this reading is selective, and that day-to-day material appears in dreams more often than Domhoff allows. 
* The theory makes its own prediction: dream emotion should rise and fall with waking distress without ever predicting improvement in it. For Natalie, that means that the dream keeps returning and her mood keeps tracking it, with the dream itself contributing nothing either way. 

===Threat simulation ===

* The second theory says the dream is rehearsal. Dreaming builds simulations of threatening situations so that the skills of noticing and escaping danger get practiced in a safe way, so what it produces is preparedness. A nightmare is the system doing its job well (Revonsuo, 2000).
* If that is right, dream threats should look like real ones and dreamers should often get away. They do neither: in a large sample of recurrent dreams, 34% contained no threat at all, fewer than 15% contained a threat that was realistic, and fewer than 20% ended in successful escape (Zadra et al., 2006). Realism matters here because the theory itself requires dream threats to resemble the dangers people actually face. 
* The supporting evidence is real but indirect. Revonsuo (2000) relies on the finding that negative emotion dominates dreams and that aggression is the most common social interaction. But dreamers tend to rate their own dreams as mostly positive while judges rate the same dreams as mostly negative, so the result depends on who is asked (Sikka et al., 2017). There is also a gap at the end of the process as the theory requires implicit learning during sleep to transfer to waking behaviour, which has not been demonstrated (Domhoff, 2019).
* For Natalie, It predicts a dream that leaves her better prepared for abandonment without leaving her any calmer, which is a poor fit for coping.

===Fear extinction ===

* The third theory says that the dream takes a frightening memory and replays it somewhere safe. This does not erase the fear: [[w:Extinction (psychology)|extinction]] builds a competing memory alongside it, which is why the fear returns if the safe memory is not maintained. A nightmare is the process failing: the replay stays frightening and no safe version gets built (Nielsen &amp; Levin, 2007).
* This model has a clinical test case to point to, since disrupted REM sleep and impaired extinction memory have been proposed to maintain [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (Pace-Schott et al., 2015).
* It also has some direct support: cathartic dreams, which move from danger to relief, were less common in patients with nightmare disorder than in healthy sleepers, and were the only kind of dream whose increase tracked falling depression scores during therapy (Perogamvros et al., 2025). The samples were small, so this points in the right direction without settling anything. 
* Its weakness is logical: a good dream and a nightmare are attributed to the same process, with success and failure identified afterwards by the very result they were meant to explain. As stated, no next-morning finding can count against the theory. It also has nothing to say about the roughly one third of dreams that are positive (Weiss, 2007). The theory only becomes testable once a successful dream can be recognised from its content in advance, which is what the cathartic dreams category is trying to do. 
* On this account Natalie's dream is a replay that never reaches safety. 

===Mood regulation ===

* Dream emotion is proposed to shift from negative toward positive across the night, so the memory arrives in the morning with less charge attached (Cartwright et al., 1998). This is the account closest to the everyday idea that dreams work through problems, and it makes the clearest predictions: after a negative dream, mood should be less negative, reactivity to negative material should be weaker, and emotion regulation should be easier (Sikka et al., 2022). 
* The overnight shift has been tested directly and did not appear. Waking people repeatedly throughout the night, Palmieri et al. (2025) found that emotional intensity in dreams increased as the night went on, and the increase was not driven by REM awakenings. The sample was small, but this does test the core claim instead of using a stand in for it. 
* Two findings do support the theory, and both are correlational. Dreamers who rated their dreams more positively than an independent judge reported better morning mood (Barbeau et al., 2022), and people who reported more fear in dreams showed weaker fear-related brain responses while awake (Sterpenich et al., 2020). Neither design can show which way the arrow points, since someone already coping well might dream more gently than and feel better for the same underlying reason. 
* The theory predicts that Natalie's dream should be losing intensity week by week, which it is not. 

==Dream emotion and waking coping ==
'''''Focus question: '''What does dream emotion predict for waking coping?''   

* Coping is what a person does while awake to manage stress (Folkman &amp; Moskowitz, 2004). For dreaming to contribute to it, dream emotion has to predict something in that waking process. 
* One objection is that a sleeping person cannot be trying to cope, but emotion regulation doesn't have to be conscious or deliberate to count (Gross, 2015).
* Coping breaks into three increasingly demanding tests: does dream emotion change how a person eels, how well they manage emotion, and whether they adapt to a real stressor? The answer differs for each, which is why this literature looks more contradictory than it is.

===Does dream emotion change how people feel? ===

*One five day study in 40 adults tested all three mood regulation predictions at once. The first result was that more negative dream emotion predicted a more negative morning (Sikka et al., 2022). 
*The largest study agrees: across 4,715 days from 536 adults, frightening dreams were followed by about 7% more negative mood the next morning (Baber et al., 2026).
*But which dream is counted changes the answer. Across up to 21 mornings in 191 adults, ordinary dreams were followed by less negative emotion the next day, and so were nightmares, against the authors own prediction. Bad dreams changed nothing, and bad dreams plus nightmares in the same night were followed by more (Tousignant et al., 2022). 
*Dreams high in both fear and joy predicted mornings with no negative mood at all, and more joy in dreams predicted more positive mood (Baber et al., 2026). Roughly a third of dreams are positive (Weiss, 2007), yet a theory about mood improving has mostly been tested by measuring only what makes mood worse. 

===Does dream emotion change how well people manage emotion? ===

* Feeling better is not the same as coping better. Coping is about handling what comes next, so the harder test is whether dream emotion predicts how a person responds to something upsetting. 
* Tested directly, it does not. In the same five day study, dream emotion had no link to reactivity to upsetting images or to the ability to calm those reactions down, and Bayesian analysis supported both null results (Sikka et al., 2022).
* This is the strongest single result against the mood regulation account, because it is evidence for no relationship rather than a failure to find one. 
* This appears to flip when people are compared with each other instead of with themselves: those who had more frightening dreams overall regulated emotion better while awake (Baber et al., 2026). 
* These are different questions, since comparing a person with themselves asks whether last nights dream helped, and comparing people asks what sort of person has fearful dreams, so only the first tests the theory. 
* Much of the apparent disagreement in this literature comes from between-person findings being read as answers to a within-person question. 

===Does dream emotion predict adapting to a real loss? ===

* Coping is defined against a stressor, so the strongest test is where there is a real one. 
* Over five months of REM awakenings in 20 depressed and 10 non-depressed adults going through a divorce, those still depressed at follow-up did dream about their ex partner, but the dreams had no emotion in them and no links to other memories. Those who recovered had emotional dreams about the same person, so it was the emotion in the dream rather than its subject that tracked recovery (Cartwright et al., 2006). 
* The same pattern appears after a death: in 77 adults with complicated grief, dreams contained far less negative emotion than published norms, 21% against 80% of dreams in women (Germain et a;. 2013). 
* The direction therefore reverses at this level. That is not a contradiction of the previous subsection, because both the outcome and the people have changed. Emotion in a dream predicts a worse morning day to day, but tracks the people who are adapting when there is a genuine loss to adapt to. 
* For Natalie, feeling the loss in the dream is a better sign than dreaming about it with no emotion, what is wrong is that the feeling is not fading. 

=== What coping research is still missing ===

* All three tests above use mood, reactivity, or recovery but none measure what a person actually does to manage stress, which is what coping means (Folkman &amp; Moskowitz, 2004). 
* The one study to measure both found that people who used positive reappraisal while awake had more problem-solving in their dreams, but that those who dreamt about the stressor reported less problem solving awake (Delorme et al., 2002). It used 22 dreamers and a mild stressor, and it is close to the whole literature connecting dreams to waking coping behaviour. 
* None of this can settle whether the dream causes anything, because no study here could change a dream, the next section does. 
&lt;quiz display=simple&gt;
{In the diary studies described above, a more negative dream was typically followed the next morning by:
|type=&quot;()&quot;}
- Better mood, as mood regulation theory predicts
|| Not quite. This is what the theory predicts, but it is not what was found.
+ Worse mood
|| Correct. Negative dream affect predicted a more negative next-day mood (Sikka et al., 2022; Baber et al., 2026).
- No detectable change in mood
|| Not quite. There was a reliable effect, but it ran in the opposite direction to the theory.
&lt;/quiz&gt;

==Using dreams to improve coping==
'''''Focus question: '''How can dreams be targeted to improve coping?''

* The previous section found that dream emotion tarcks how well someone is coping without clear evidence that it drives it, which may be a real limit or only a limit of the method. 
* This section reverse the direction of the measurement, instead of recording a dream and looking forward to the morning, it changes something while the person is awake and then looks back at the dream. That is the only order that can show cause. 

{{RoundBoxTop|theme=11}}
'''Case study: part 2'''

Six months on, the dream still hasn't stopped. Natalie has started staying up until she's too tired to dream, and she's tired all the time. Her psychologist suggests writing the dream down and changing what happens: this time, the person turns around, or Natalie stops calling out and walks the other way. She rehearses the new version of the dream for a few minutes every night. Within weeks the dream starts to come less often, and when it does, it bothers her less.     
{{RoundBoxBottom}}

===Imagery rehearsal therapy ===

* Imagery rehearsal therapy (IRT) means writing down a recurring nightmare, deliberately changing how it ends, and rehearsing the new version while awake for 10 to 20 minutes a day (Morgenthaler et al., 2018).
* The logic follows fear extinction: rehearsing a new ending does while awake what the theory says a healthy dream does at night.
* It is the only treatment the American Academy of Sleep Medicine recommends for nightmare disorder (Morgenthaler et al., 2018), and across 29 randomised trials it was one of the only two treatments to beat control conditions (Y. Zhang et al., 2022).
* The large effects are uncontrolled: improvements in nightmares, sleep and PTSD held at 6 and 12 months across 13 studies but shrank to a small effect in the one trial that compared it against another active treatment (Casement &amp; Swanson, 2012), and two later trials found no advantage over other sleep treatments in veterans with PTSD (Cook et al., 2010; Harb et al., 2019). 
* Why IRT works remains disputed: a systematic review of the proposed mechanisms found none established (Rousseau &amp; Belleville, 2018), and 30% of patients did not respond at all (Schwartz et al., 2022). 
* Implication: If IRT works through waking rehearsal and changed expectations, the dream is the target of treatment rather than the agent of change.

=== Manipulating dream content: the missing method ===

* A causal claim about dreaming requires manipulating dream content, which nothing in the previous sections does. Targeted dream incubation repeats a chosen theme at sleep onset: 67% of prompted awakenings produced dream reports containing the theme, against 3% of unprompted awakenings (Haar Horowitz et al., 2020). Dream content is therefore controllable. 
* But this is a methods demonstration rather than a treatment, even successful incubation cannot show whether the experience of dreaming matters or merely accompanies the work being done (Haar Horowitz et al., 2020). 
* Working the other way, an improved capacity for affect regulation within dreams accompanied improvement in personality functioning during psychotherapy, so dream content may index progress as well as be a target of it (Kempe et al., 2024).
* What would settle the question is an experiment that manipulates dream emotion under real distress and measures waking coping rather than nightmare frequency. Nothing in this literature does that yet. 

==Conclusion==
* Dreams like Natalie's recur because the emotional charge attached to a memory has not faded overnight. That is the problem this chapter set out to explain. 
* The short answer to the subtitle is that REM dreams reflect emotional processing more reliably than they perform it. Dream emotion behaves more like a thermometer than a heater: it registers how much distress someone is carrying, but there is little evidence that it is what brings the distress down. 
* Sleep and dreaming are not the same thing. Sleep makes emotional memories more likely to be remembered without reliably making them feel less bad (Lipinska et al., 2019), and dreaming continues when REM sleep is supressed (Solms, 2000). Only two studies have tested whether the dream itself adds anything (Lara-Carrasco et al., 2009; J. Zhang et al., 2024).
* Four theories disagree about what a dream is for and they can be separated by what each predicts about the following morning (see Table 1).
* The prediction that matters most for coping is the mood regulation one and it fails. Dream emotion predicts a worse next morning rather than a better one, and predicts nothing about reactivity or about the ability to regulate emotion (Sikka et al., 2022; Baber et al., 2026), so it looks more like a sign of how well someone is coping than the thing doing the coping.
* Coping itself has almost never been measured, every finding above uses mood, reactivity or clinical recovery as a stand in for what a person actually does to manage stress (Folkman &amp; Moskowitz, 2004; Delorme et al., 2002). 
* Dreams can still be changed on purpose: rewriting a recurring nightmare reduces both how often it happens and how much distress it causes (Morgenthaler et al., 2018), though whether the rewriting is the active ingredient is unresolved (Cook et al., 2010; Harb et al., 2019; Schwartz et al., 2022).
* Settling the wider question means controlling dream content rather than only recording it (Haar Horowitz et al., 2020; Schwartz et al., 2022), and measuring coping rather than nightmare frequency. 
* Natalie's dream did not repair itself, and it did not need to. What changed it was practising a new ending while awake.

{{RoundBoxTop|theme=11}}
'''Take-home messages'''

* A bad dream is not a sign that emotional processing is doing its job, dream emotion tracks distress rather than relieving it. 
* Dreaming may still add something, but the evidence is thin and comes from a handful of studies that could not assign anyone to dream.  
*&quot;Sleep on it&quot; is reasonable advice for retaining a memory, but not a reliable way to feel better about it.  
  
{{RoundBoxBottom}}

==See also==

* [[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020)
* [[Motivation and emotion/Book/2025/Dreams and emotional problem-solving|Dreams and emotional problem-solving]] (Book chapter, 2025)
* [[w:Emotional self-regulation|Emotional self-regulation]] (Wikipedia)
* [[w:Nightmare disorder|Nightmare disorder]] (Wikipedia)
* [[Motivation and emotion/Book/2020/Nightmares and emotion|Nightmares and emotion]] (Book chapter, 2020)

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{{Hanging indent|1=
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Scarpelli, S., Alfonsi, V., Gorgoni, M., &amp; De Gennaro, L. (2022). What about dreams? State of the art and open questions. ''Journal of Sleep Research, 31''(4), e13609. https://doi.org/10.1111/jsr.13609

Schredl, M. (2017). Theorizing about the continuity between waking and dreaming: Comment on Domhoff (2017). ''Dreaming, 27''(4), 351–359. https://doi.org/10.1037/drm0000062

Schwartz, S., Clerget, A., &amp; Perogamvros, L. (2022). Enhancing imagery rehearsal therapy for nightmares with targeted memory reactivation. ''Current Biology, 32''(22), 4808-4816.e4. https://doi.org/10.1016/j.cub.2022.09.032 

Sikka, P., Engelbrektsson, H., Zhang, J., &amp; Gross, J. J. (2022). Negative dream affect is associated with next-day affect level, but not with affect reactivity or affect regulation. ''Frontiers in Behavioral Neuroscience, 16'', 981289. https://doi.org/10.3389/fnbeh.2022.981289

Sikka, P., Feilhauer, D., Valli, K., &amp; Revonsuo, A. (2017). How you measure is what you get: Differences in self- and external ratings of emotional experiences in home dreams. ''The American Journal of Psychology, 130''(3), 367–384. https://doi.org/10.5406/amerjpsyc.130.3.0367

Solms, M. (2000). Dreaming and REM sleep are controlled by different brain mechanisms. ''Behavioral and Brain Sciences, 23''(6), 843–850. https://doi.org/10.1017/S0140525X00003988

Sterpenich, V., Perogamvros, L., Tononi, G., &amp; Schwartz, S. (2020). Fear in dreams and in wakefulness: Evidence for day/night affective homeostasis. ''Human Brain Mapping, 41''(3), 840–850. https://doi.org/10.1002/hbm.24843 

Tousignant, O. H., Glass, D. J., Suvak, M. K., &amp; Fireman, G. D. (2022). Nightmares and nondisturbed dreams impact daily change in negative emotion. ''Dreaming (New York, N.Y.), 32''(3),292-313. Advance online publication. https://doi.org/10.1037/drm0000213 

van der Helm, E., Yao, J., Dutt, S., Rao, V., Saletin, J. M., &amp; Walker, M. P. (2011). REM sleep depotentiates amygdala activity to previous emotional experiences. ''Current Biology, 21''(23), 2029–2032. https://doi.org/10.1016/j.cub.2011.10.052

Walker, M. P., &amp; van der Helm, E. (2009). Overnight therapy? The role of sleep in emotional brain processing. ''Psychological Bulletin, 135''(5), 731–748. https://doi.org/10.1037/a0016570

Weiss, D. S. (2007). Conundrums in a theory of disturbed dreaming: Comment on Levin and Nielsen (2007). ''Psychological Bulletin, 133''(3), 529–532. https://doi.org/10.1037/0033-2909.133.3.529

Zhang, J., Pena, A., Delano, N., Sattari, N., Shuster, A. E., Baker, F. C., Simon, K., &amp; Mednick, S. C. (2024). Evidence of an active role of dreaming in emotional memory processing shows that we dream to forget. ''Scientific Reports, 14''(1), Article 8722. https://doi.org/10.1038/s41598-024-58170-z &lt;!-- add article number --&gt;

Zhang, Y., Ren, R., Vitiello, M. V., Yang, L., Zhang, H., Shi, Y., Sanford, L. D., &amp; Tang, X. (2022). Efficacy and acceptability of psychotherapeutic and pharmacological interventions for trauma-related nightmares: A systematic review and network meta-analysis. ''Neuroscience &amp; Biobehavioral Reviews, 139'', 104717. https://doi.org/10.1016/j.neubiorev.2022.104717
}}

==External links==
* [https://www.sleepfoundation.org/nightmares/how-to-stop-having-nightmares How to prevent nightmares] (Sleep Foundation)
* [https://shows.acast.com/sleep-science-pod/episodes/ep12-sleep-dreams-and-mental-health Sleep, dreams and mental health] (The Sleep Science Pod)
* [https://www.media.mit.edu/projects/targeted-dream-incubation/overview/ Targeted dream incubation] (MIT Media Lab)
* [https://theconversation.com/the-science-of-dreams-and-nightmares-what-is-going-on-in-our-brains-while-were-sleeping-210901 The science of dreams and nightmares – what is going on in our brains while we're sleeping?] (The Conversation)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
*

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[[Category:Motivation and emotion/Book/Dreams]]
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape. They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the &quot;in the moment experience of pleasure&quot;.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour. 
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. 

=== Anticipatory and consummatory pleasure - Gard et al. (2006) ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates antipatory pleasure with the motivation process of 'wanting' 
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are strongly correlated however represent seperate constructs. 

=== Pleasure and motivated behaviour - Geanney et al. (2015) ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. 
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. 
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced.  

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. 

=== The dopamine reward system - Nasser et. al., 2017 ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation.
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. 
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward.  

=== Reward prediction and learning - Nasser et. al., 2017 ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. 
* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. 
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. 

=== Wanting versus liking - Berridge &amp; Robinson (2016) ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. 
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. 
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation.

=== Incentive salience theory - Berridge &amp; Robinson (2016) ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward.
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. 
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. 

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. 

=== Sexual desire and pleasure - Agmo &amp; Laan (2023) ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. 
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire.
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. 

=== Gambling, reward, and uncertainty - Clark et. al. (2009) ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. 
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. 
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. 

=== Substance use and addiction - Berridge &amp; Robinson (2016) ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. 
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour.
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. 

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
[[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape. They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the &quot;in the moment experience of pleasure&quot;.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour. 
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. 

=== Anticipatory and consummatory pleasure - Gard et al. (2006) ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates antipatory pleasure with the motivation process of 'wanting' 
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are strongly correlated however represent seperate constructs. 

=== Pleasure and motivated behaviour - Geanney et al. (2015) ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. 
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. 
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced.  

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. 

=== The dopamine reward system - Nasser et. al., 2017 ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation.
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. 
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward.  

=== Reward prediction and learning - Nasser et. al., 2017 ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. 
* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. 
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. 

=== Wanting versus liking - Berridge &amp; Robinson (2016) ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. 
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. 
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation.

=== Incentive salience theory - Berridge &amp; Robinson (2016) ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward.
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. 
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. 

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. 

=== Sexual desire and pleasure - Agmo &amp; Laan (2023) ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. 
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire.
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. 

=== Gambling, reward, and uncertainty - Clark et. al. (2009) ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. 
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. 
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. 

=== Substance use and addiction - Berridge &amp; Robinson (2016) ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. 
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour.
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. 

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
[[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Ågmo, A., &amp; Laan, E. (2022). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape. They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the &quot;in the moment experience of pleasure&quot;.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour. 
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. 

=== Anticipatory and consummatory pleasure - Gard et al. (2006) ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates antipatory pleasure with the motivation process of 'wanting' 
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are strongly correlated however represent seperate constructs. 

=== Pleasure and motivated behaviour - Geanney et al. (2015) ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. 
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. 
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced.  

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. 

=== The dopamine reward system - Nasser et. al., 2017 ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation.
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. 
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward.  

=== Reward prediction and learning - Nasser et. al., 2017 ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. 
* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. 
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. 

=== Wanting versus liking - Berridge &amp; Robinson (2016) ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. 
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. 
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation.

=== Incentive salience theory - Berridge &amp; Robinson (2016) ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward.
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. 
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. 

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. 

=== Sexual desire and pleasure - Agmo &amp; Laan (2023) ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. 
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire.
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. 

=== Gambling, reward, and uncertainty - Clark et. al. (2009) ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. 
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. 
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. 

=== Substance use and addiction - Berridge &amp; Robinson (2016) ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. 
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour.
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. 

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
[[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1= 
Ågmo, A., &amp; Laan, E. (2022). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 
}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape. They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the &quot;in the moment experience of pleasure&quot;.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour. 
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. 

=== Anticipatory and consummatory pleasure - Gard et al. (2006) ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates antipatory pleasure with the motivation process of 'wanting' 
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are strongly correlated however represent seperate constructs. 

=== Pleasure and motivated behaviour - Geanney et al. (2015) ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. 
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. 
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced.  

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. 

=== The dopamine reward system - Nasser et. al., 2017 ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation.
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. 
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward.  

=== Reward prediction and learning - Nasser et. al., 2017 ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. 
* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. 
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. 

=== Wanting versus liking - Berridge &amp; Robinson (2016) ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. 
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. 
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation.

=== Incentive salience theory - Berridge &amp; Robinson (2016) ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward.
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. 
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. 

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. 

=== Sexual desire and pleasure - Agmo &amp; Laan (2023) ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. 
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire.
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. 

=== Gambling, reward, and uncertainty - Clark et. al. (2009) ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. 
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. 
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. 

=== Substance use and addiction - Berridge &amp; Robinson (2016) ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. 
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour.
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. 

=== Quiz ===
&lt;quiz display=&quot;simple&quot;&gt;

{What is anticipatory pleasure?
|type=&quot;()&quot;}

- Experienced pleasure only AFTER receiving a reward
+ Pleasure associated with expecting a future reward
- The lack of motivation for a reward
- Pleasure that is only experienced while experiencing a pleasurable reward

{When does a positive reward prediction error occur?
|type=&quot;()&quot;}

+ When a reward is greater than expected 
- When a reward is equally to what was expected 
- When a reward is less than what was expected 
- When there is no reward granted 

{According to incentive salience theory, what does &quot;wanting&quot; refer to?
|type=&quot;()&quot;}

- The pleasure experinced when receiving a reward 
- The absensce of reward anticipation 
- Having a fear of rewarding stimuli 
+ The motivation to pursue a reward 

{Which example best demonstrates &quot;wanting&quot; without &quot;liking&quot;?
|type=&quot;()&quot;}

- Enjoying a reward you originally weren't anticipating you would enjoy
- Avoiding the reward as you don't anticipate you will enjoy it 
+ Strongly cravings a substance despite experiencing little pleasure from it 
- Receiving an pleasurable unexpected reward 

&lt;/quiz&gt;


==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
[[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1= 
Ågmo, A., &amp; Laan, E. (2022). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 
}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape. They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the &quot;in the moment experience of pleasure&quot;.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour. 
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. 

=== Anticipatory and consummatory pleasure - Gard et al. (2006) ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates antipatory pleasure with the motivation process of 'wanting' 
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are strongly correlated however represent seperate constructs. 

=== Pleasure and motivated behaviour - Geanney et al. (2015) ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. 
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. 
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced.  

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. 

=== The dopamine reward system - Nasser et. al., 2017 ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation.
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. 
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward.  

=== Reward prediction and learning - Nasser et. al., 2017 ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. 
* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. 
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. 

=== Wanting versus liking - Berridge &amp; Robinson (2016) ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. 
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. 
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation.

=== Incentive salience theory - Berridge &amp; Robinson (2016) ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward.
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. 
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. 

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. 

=== Sexual desire and pleasure - Agmo &amp; Laan (2023) ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. 
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire.
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. 

=== Gambling, reward, and uncertainty - Clark et. al. (2009) ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. 
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. 
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. 

=== Substance use and addiction - Berridge &amp; Robinson (2016) ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. 
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour.
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. 

=== Quiz ===
&lt;quiz display=&quot;simple&quot;&gt;

{What is anticipatory pleasure?
|type=&quot;()&quot;}

- Experienced pleasure only AFTER receiving a reward
+ Pleasure associated with expecting a future reward
- The lack of motivation for a reward
- Pleasure that is only experienced while experiencing a pleasurable reward

{When does a positive reward prediction error occur?
|type=&quot;()&quot;}

+ When a reward is greater than expected 
- When a reward is equally to what was expected 
- When a reward is less than what was expected 
- When there is no reward granted 

{According to incentive salience theory, what does &quot;wanting&quot; refer to?
|type=&quot;()&quot;}

- The pleasure experinced when receiving a reward 
- The absensce of reward anticipation 
- Having a fear of rewarding stimuli 
+ The motivation to pursue a reward 

{Which example best demonstrates &quot;wanting&quot; without &quot;liking&quot;?
|type=&quot;()&quot;}

- Enjoying a reward you originally weren't anticipating you would enjoy
- Avoiding the reward as you don't anticipate you will enjoy it 
+ Strongly cravings a substance despite experiencing little pleasure from it 
- Receiving an pleasurable unexpected reward 

&lt;/quiz&gt;


==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure
* Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[Motivation and emotion/Book/2026/Reward prediction error|Reward Prediction Error]] 
* [[Motivation and emotion/Book/2025/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine (2025)]]
* [https://www.healthdirect.gov.au/dopamine?trk=public_post_comment-text&amp;utm Dopamine] 
* [https://www.healthdirect.gov.au/gambling-addiction?utm Gambling Addiction] 

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1= 
Ågmo, A., &amp; Laan, E. (2022). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 
}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape (Figure One). They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the &quot;in the moment experience of pleasure&quot;.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour. 
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. 

=== Anticipatory and consummatory pleasure - Gard et al. (2006) ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates antipatory pleasure with the motivation process of 'wanting' 
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are strongly correlated however represent seperate constructs. 

=== Pleasure and motivated behaviour - Geanney et al. (2015) ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. 
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. 
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced.  

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. 

=== The dopamine reward system - Nasser et. al., 2017 ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation.
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. 
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward.  

=== Reward prediction and learning - Nasser et. al., 2017 ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. 
* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. 
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. 

=== Wanting versus liking - Berridge &amp; Robinson (2016) ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. 
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. 
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation.

=== Incentive salience theory - Berridge &amp; Robinson (2016) ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward.
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. 
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. 

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. 

=== Sexual desire and pleasure - Agmo &amp; Laan (2023) ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. 
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire.
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. 

=== Gambling, reward, and uncertainty - Clark et. al. (2009) ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. 
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. 
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. 

=== Substance use and addiction - Berridge &amp; Robinson (2016) ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. 
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour.
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. 

=== Quiz ===
&lt;quiz display=&quot;simple&quot;&gt;

{What is anticipatory pleasure?
|type=&quot;()&quot;}

- Experienced pleasure only AFTER receiving a reward
+ Pleasure associated with expecting a future reward
- The lack of motivation for a reward
- Pleasure that is only experienced while experiencing a pleasurable reward

{When does a positive reward prediction error occur?
|type=&quot;()&quot;}

+ When a reward is greater than expected 
- When a reward is equally to what was expected 
- When a reward is less than what was expected 
- When there is no reward granted 

{According to incentive salience theory, what does &quot;wanting&quot; refer to?
|type=&quot;()&quot;}

- The pleasure experinced when receiving a reward 
- The absensce of reward anticipation 
- Having a fear of rewarding stimuli 
+ The motivation to pursue a reward 

{Which example best demonstrates &quot;wanting&quot; without &quot;liking&quot;?
|type=&quot;()&quot;}

- Enjoying a reward you originally weren't anticipating you would enjoy
- Avoiding the reward as you don't anticipate you will enjoy it 
+ Strongly cravings a substance despite experiencing little pleasure from it 
- Receiving an pleasurable unexpected reward 

&lt;/quiz&gt;
==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

==See also==
* [[Motivation and emotion/Book/2026/Reward prediction error|Reward Prediction Error]] 
* [[Motivation and emotion/Book/2025/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine (2025)]]

==References==
{{Hanging indent|1= 
Ågmo, A., &amp; Laan, E. (2022). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 
}}

==External links==

*[https://www.healthdirect.gov.au/dopamine?trk=public_post_comment-text&amp;utm Dopamine]
* [https://www.healthdirect.gov.au/gambling-addiction?utm Gambling Addiction]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape (Figure One). They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure refers to epxerienced when thinking about or expecting a future rewarding experience. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of wanting. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the in the moment experience of pleasure.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure and the participants willingness to expand effort for monetary rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling. 
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting which cues signal future rewards. Nasser et al. (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour.
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) distinguish between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it. 

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. (Gard et al. (2006)

=== Anticipatory and consummatory pleasure ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates anticipatory pleasure with the motivation process of 'wanting'
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are moderately positively correlated however represent seperate constructs.

=== Pleasure and motivated behaviour ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. Geanney et al. (2015)
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. Geanney et al. (2015)
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced. Geanney et al. (2015)

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. (Nasser et. al., 2017) 

=== The dopamine reward system ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation. (Nasser et. al., 2017)
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. (Nasser et. al., 2017)
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward. (Nasser et. al., 2017)

=== Reward prediction and learning ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. (Nasser et. al., 2017)

* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. (Nasser et. al., 2017)
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. (Nasser et. al., 2017) 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. (Berridge &amp; Robinson 20016) 

=== Wanting versus liking ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. (Berridge &amp; Robinson (2016)
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. (Berridge &amp; Robinson (2016)
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation. (Berridge &amp; Robinson (2016)

=== Incentive salience theory ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward. (Berridge &amp; Robinson (2016)
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. (Berridge &amp; Robinson (2016)
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. (Berridge &amp; Robinson (2016)

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. (Agmo &amp; Laan, 2023)

=== Sexual desire and pleasure ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. (Agmo &amp; Laan (2023)
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire. (Agmo &amp; Laan (2023)
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. (Agmo &amp; Laan (2023)

=== Gambling, reward, and uncertainty ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. (Clark et. al. (2009)
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. (Clark et. al. (2009)
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. (Clark et. al. (2009)

=== Substance use and addiction ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. ( Berridge &amp; Robinson (2016)
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour. (Berridge &amp; Robinson (2016)
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. (Berridge &amp; Robinson (2016)

=== Quiz ===
&lt;quiz display=&quot;simple&quot;&gt;

{What is anticipatory pleasure?
|type=&quot;()&quot;}

- Experienced pleasure only AFTER receiving a reward
+ Pleasure associated with expecting a future reward
- The lack of motivation for a reward
- Pleasure that is only experienced while experiencing a pleasurable reward

{When does a positive reward prediction error occur?
|type=&quot;()&quot;}

+ When a reward is greater than expected 
- When a reward is equally to what was expected 
- When a reward is less than what was expected 
- When there is no reward granted 

{According to incentive salience theory, what does &quot;wanting&quot; refer to?
|type=&quot;()&quot;}

- The pleasure experinced when receiving a reward 
- The absensce of reward anticipation 
- Having a fear of rewarding stimuli 
+ The motivation to pursue a reward 

{Which example best demonstrates &quot;wanting&quot; without &quot;liking&quot;?
|type=&quot;()&quot;}

- Enjoying a reward you originally weren't anticipating you would enjoy
- Avoiding the reward as you don't anticipate you will enjoy it 
+ Strongly cravings a substance despite experiencing little pleasure from it 
- Receiving an pleasurable unexpected reward 

&lt;/quiz&gt;
==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

==See also==
* [[Motivation and emotion/Book/2026/Reward prediction error|Reward Prediction Error]] 
* [[Motivation and emotion/Book/2025/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine (2025)]]

==References==
{{Hanging indent|1= 
Ågmo, A., &amp; Laan, E. (2023). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 
}}

==External links==

*[https://www.healthdirect.gov.au/dopamine?trk=public_post_comment-text&amp;utm Dopamine]
* [https://www.healthdirect.gov.au/gambling-addiction?utm Gambling Addiction]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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==Overview ==
{{RoundBoxTop|theme=3}}
;Scenario - Wanting without liking 

[[File:Cartoon describing the vaporization process of e-cigarette.webp|thumb|right|120px|Figure 1. Illustration of the vaporisation process of an electronic cigarette.]]

A university student is up late studying for their final exam, they're under considerable stress and are experiencing a strong desire to use their vape (Figure One). They anticipate that using their vape will feel satisfying and rewarding as they can not focus on anything else. Eventually they use their vape, however the experience is disappointing and uncomfortable. The taste is unpleasant and they now feel nauseous, the expected satisfaction is breif. Despite this, the craving later returns. Why can the anticipation of a experience create such a strong motivation when the experience itself involves little pleasure?  
{{RoundBoxBottom}}
*'''What is pleasure anticipation?''' - Anticipatory pleasure is epxerienced when thinking about or expecting a future reward. Gard et al. (2006) described anticipatory pleasure as the pleasure experienced in anticipation of an enjoyable stimuli and associates it with the feeling of 'wanting'. While consummatory pleasure is the pleasure experinerenced when the rewarding stimuli is actually taking place, often described as the in the moment experience of pleasure.
*'''How does pleasure anticipation motivate behaviour? -''' Anticipating pleasure can influence motivated behaviour by increasing a persons willingness to put effort into obtaining a potential future reward. Geaney et al. (2015) exmained aticipatory pleasure along with participants willingness to expand their efforts for rewards. In doing this it was found that higher aticipary pleasure was associated with more effortful choices for potially greater rewards. This now suggests that pleasure anticipation can influence reward seeking motivation, rather than simply being a positive feeling.
*'''What role does dopamine play in reward anticipation? -''' Dopamine plays a key role in reward anticipation by assisting the brain in learning and predicting, which cues signal future rewards. Nasser et al. (2017) describes dopamine reward prediction error signalling as reflecting differences between expected and received rewards. This allows reward expectations to be updated through learning. This process helps the brain learn which cues are associated with rewards, which can motivate future reward seeking behaviour.
*'''What is the difference between wanting vs liking?''' - Berridge and Robinson (2016) study distinguishes between the two components of reward (Wanting vs Liking). The literature explains that incentive salience (wanting), is a form of motivation associated with mesolimbic dopamine systems, compared to 'liking' which refers to the pleasurable feeling experienced when a reward is consumed. Crucially, Berridge and Robinson state that the brain mechanisms responsible for 'wanting' and 'liking' are seperate, meaning a person can strongly want a reward without necessarily experiencing the same level of pleasure when they actually receive it.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does anticipating pleasure influence motivation and behaviour? 
* What role does dopamine play in reward anticipation, prediction and learning? 
* How can incentive salience explain the difference between 'wanting' and 'liking' a reward? 
* How does pleasure anticipation influence behaviours such as sexual desire, gambling and substance use? 
{{RoundBoxBottom}}

==Pleasure anticipation and motivation ==

* Pleasure anticipation is a key component of reward based motivation.
* Experiencing reward can involve anticipatory pleasure even prior to receiving a reward and consummatory pleasure while experiencing the reward.   
* Expectations of future pleasure can motivate individuals to engage in behaviours directed towards obtaining a potential reward. (Gard et al. (2006)

=== Anticipatory and consummatory pleasure ===

* Anticipatory pleasure is referring too the pleasure experincned when thinking about anticipating a future reward or pleasurable feeling. Gard et al. (2006) associates anticipatory pleasure with the motivation process of 'wanting'
* Consummatory pleasure refers to the pleasure experienced while the rewarding event is actually happening. Gard (2006) refers to this as an 'in the moment' experience of pleasure. 
* Anticipatory and consummatory pleasure are related but are two seperate experiences. Gard (2006) found that the two components are moderately positively correlated however represent seperate constructs.

=== Pleasure and motivated behaviour ===

* Anticipating pleasure can motivate reward seeking behaviour. The expectation of a future rewarding experience can increase the motivation and willingness of an individual to put more effort into obtaining that suspected reward. Geanney et al. (2015)
* Anticipatory pleasure predicted effort towards reward. It was found that higher anticipatory pleasure was linked to greater effort for rewards, especially when the predicted chances of the reward was low. Geanney et al. (2015)
* These findings suggest that expected pleasure can influence motivated behaviour prior to a reward being experienced. Geanney et al. (2015)

== Dopamine and reward anticipation ==

* Dopamine plays an key role in reward anticipation, learning and motivated behaviour. 
* Dopamine activity can respond to cues that predict future rewards, allowing these cues to gain motivation significance. 
* Errors in reward prediction can help individuals learn differences between expected and experienced rewards. (Nasser et. al., 2017) 

=== The dopamine reward system ===

* Dopamine is an important neurotransmitter involved in the brains connected reward systems in the limbic regions and contributes to reward related learning and motivation. (Nasser et. al., 2017)
* Dopamine neurons in the midbrain respond to unexpected rewards. As learning takes place, dopamine activity can begin responding to the cue that signals the upcoming reward. (Nasser et. al., 2017)
* Reward predicting cues can then therefore become motivationally significant. Once a cue becomes associated with a reward it can then encourage future behaviour in regards to obtaining a reward. (Nasser et. al., 2017)

=== Reward prediction and learning ===

* Reward prediction error occurs when there is a difference between the expected reward and the reward thats actually received. (Nasser et. al., 2017)

* When a reward is greater then originally predicted that is described as the positive prediction error, while when a reward is less than predicted, a negative prediction error is produced. (Nasser et. al., 2017)
* Prediction errors support learning by updating an individuals predictions of future rewards. Overtime, this can allow cues to become associated with particular rewards and either positively or negatively influence future decisions. (Nasser et. al., 2017) 

== Wanting, liking and incentive salience ==

* Reward involves seperate processes of 'wanting' and 'liking', which often influenced reward seeking behaviour. 
* 'Wanting' refers to the motivation to pursue a reward, whereas 'liking' refers to the pleasure experienced when the reward is received. 
* Incentive salience can influence motivational value of rewarding cues, meaning an indivdual may strongly 'want' a reward without necessarily experinecing that same level of 'liking' the reward. (Berridge &amp; Robinson 20016) 

=== Wanting versus liking ===

* 'Wanting' and 'liking' are two seperate components of reward. 'Wanting' refers to the motivation to obtain a reward, while 'liking' refers to the pleasure felt once the reward is experienced. (Berridge &amp; Robinson (2016)
* The systems in the brain involved in 'wanting' and 'liking' can operate separately. Individuals can strongly desire something even when the actual experienced provides little pleasure. (Berridge &amp; Robinson (2016)
* Dopamine is more strongly linking with wanting than liking, challenging the idea that dopamine simply produced pleasure and rather highlights its role in reward motivation. (Berridge &amp; Robinson (2016)

=== Incentive salience theory ===

* Incentive salience theory explains how reward related cues can strongly motivate behaviour. Cues associated with reward can trigger 'wanting' for the reward. (Berridge &amp; Robinson (2016)
* Mesolimbic dopamine systems play a key role in incentive salience through increasing the motivational value of these reward related cues. (Berridge &amp; Robinson (2016)
* Incentive salience can often help explain why the feeling of 'wanting' may continue or increase while 'liking' decrease. This can lead to an individual to remain highly motivated to seek a reward despite receiving limited pleasure from it. (Berridge &amp; Robinson (2016)

== Pleasure anticipation and human behaviour ==

* Pleasure anticipation can motivate individuals to seek rewarding experiences. 
* Reward related cues can increase motivation when pleasure is expected. 
* Sexual desire, gambling and substance abuse demonstrate how anticipated rewards can influence motivation and reward seeking behaviour. (Agmo &amp; Laan, 2023)

=== Sexual desire and pleasure ===

* Sexual desire can be seen as a form of incentive motivation. Sexual cues can active motivation and encourage individuals to approach or seek sexual experiences. (Agmo &amp; Laan (2023)
* Expecting sexual pleasure can make sexual cues more motivating. Previous pleasurable sexual experiences can make related cues more attractive and increase future sexual desire. (Agmo &amp; Laan (2023)
* Sexual motivation can often occur prior to experienced pleasure. Individuals may seek sexual experienced because they expect them to be reward, even before they feel pleasure. (Agmo &amp; Laan (2023)

=== Gambling, reward, and uncertainty ===

* The uncertainty of whether a reward will be granted may increase an individuals motivation to gamble. Unpredictable rewards may make gambling related cues more motivating, further inspiring one to seek rewards. (Clark et. al. (2009)
* This study found that 'near misses' in gambling increased participants motivation to continue playing, despite the 'near misses' being experienced as unpleasant. (Clark et. al. (2009)
* Near misses illustrate how 'wanting' can occur without 'liking'. Individuals kay contine to feel motivation to gamble even when they do not receive an immediate reward of enjoy the outcome. (Clark et. al. (2009)

=== Substance use and addiction ===

* Substance use further demonstrates how 'wanting' and 'liking' can become seperate. An individual may develop a strong motivation to use a substance even when the pleasurable effects have decreased or do not exist. ( Berridge &amp; Robinson (2016)
* Repeated drug use can sensitise brain systems involved in incentive motivation. This can lead to drug related cues becoming more powerful leading to a trigger in 'wanting' and drug seeking behaviour. (Berridge &amp; Robinson (2016)
* Drug related cues can therefore continue to trigger strong cravings, even when the experience of the substance is no longer considered pleasurable. This can explain how excessive 'wanting' can contribute to persistent substance seeking and relapse. (Berridge &amp; Robinson (2016)

=== Quiz ===
&lt;quiz display=&quot;simple&quot;&gt;

{What is anticipatory pleasure?
|type=&quot;()&quot;}

- Experienced pleasure only AFTER receiving a reward
+ Pleasure associated with expecting a future reward
- The lack of motivation for a reward
- Pleasure that is only experienced while experiencing a pleasurable reward

{When does a positive reward prediction error occur?
|type=&quot;()&quot;}

+ When a reward is greater than expected 
- When a reward is equally to what was expected 
- When a reward is less than what was expected 
- When there is no reward granted 

{According to incentive salience theory, what does &quot;wanting&quot; refer to?
|type=&quot;()&quot;}

- The pleasure experinced when receiving a reward 
- The absensce of reward anticipation 
- Having a fear of rewarding stimuli 
+ The motivation to pursue a reward 

{Which example best demonstrates &quot;wanting&quot; without &quot;liking&quot;?
|type=&quot;()&quot;}

- Enjoying a reward you originally weren't anticipating you would enjoy
- Avoiding the reward as you don't anticipate you will enjoy it 
+ Strongly cravings a substance despite experiencing little pleasure from it 
- Receiving an pleasurable unexpected reward 

&lt;/quiz&gt;
==Conclusion==
* Pleasure anticipation can generate motivation prior to actually experiencing the reward. Expectations of future pleasure can encourage reward seeking behaviours and increase an individuals willingness to chase a reward. 
* Dopamine contributes to reward anticipation through reward prediction, learning and incentive motivation. Reward related cues can become significant and often influence behaviour towards obtaining rewards. 
* Anticipated and experienced pleasure do not always correspond. There is a key distinction between 'wanting' and 'liking' which helps explain why individuals may remain heavily motivated to pursue pleasurable experineces such as sexual pleasure, gambling and substance use, even if the reward only provide limited pleasure. 

==See also==
* [[Motivation and emotion/Book/2026/Reward prediction error|Reward Prediction Error]] 
* [[Motivation and emotion/Book/2025/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine (2025)]]

==References==
{{Hanging indent|1= 
Ågmo, A., &amp; Laan, E. (2023). The sexual incentive motivation model and its clinical applications. ''The Journal of Sex Research'', ''60''(7), 969–988. https://doi.org/10.1080/00224499.2022.2134978

Berridge,     K. C., &amp; Robinson, T. E. (2016). Liking, wanting, and the     incentive-sensitization theory of addiction. ''The American Psychologist'', ''71''(8),     670–679. PubMed Central. [https://pmc.ncbi.nlm.nih.gov/articles/PMC5171207/?utm https://doi.org/10.1037/amp0000059]

Clark,     L., Lawrence, A. J., Astley-Jones, F., &amp; Gray, N. (2009). Gambling     near-misses enhance motivation to gamble and recruit win-related brain     circuitry. ''Neuron'', ''61''(3),     481–490. PubMed Central. https://doi.org/10.1016/j.neuron.2008.12.031

Gard, D.     E., Gard, M. G., Kring, A. M., &amp; John, O. P. (2006). Anticipatory and     consummatory components of the experience of pleasure: A scale development     study. ''Journal of Research in Personality'', ''40''(6), 1086–1102. https://doi.org/10.1016/j.jrp.2005.11.001

Geaney,     J. T., Treadway, M. T., &amp; Smillie, L. D. (2015). Trait anticipatory     pleasure predicts effort expenditure for reward. ''PLOS ONE'', ''10''(6),     e0131357. https://doi.org/10.1371/journal.pone.0131357 

Nasser,     H. M., Calu, D. J., Schoenbaum, G., &amp; Sharpe, M. J. (2017). The     dopamine prediction error: Contributions to associative models of reward     learning. ''Frontiers in Psychology'', ''8''. https://doi.org/10.3389/fpsyg.2017.00244 
}}

==External links==

*[https://www.healthdirect.gov.au/dopamine?trk=public_post_comment-text&amp;utm Dopamine]
* [https://www.healthdirect.gov.au/gambling-addiction?utm Gambling Addiction]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]


[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Neurotransmitters/Dopamine]]
[[Category:Motivation and emotion/Book/Pleasure]]</text>
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The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===[[/Glossary/|Glossary]]===

===Arithmetic===

===Pre-Algebra===

===Algebra===
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[/Wishlist/|Wishlist]] page if you must.</text>
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The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===[[/Glossary/|Glossary]]===

===Arithmetic===

===Pre-Algebra===

===Algebra===
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or iteratively edit and save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[/Wishlist/|Wishlist]] page where you can add text, but not a blank page, if you must. You can also request additions to existing pages, like including irrational exponents and examples in the existing [[/Exponents/|Exponents]] page. You can also contribute specific exercises and examples, like real-world applications that could interest students, and request someone translate them into TeX.</text>
      <sha1>nybvios63hccjoodhdmgun7w78z5pd3</sha1>
    </revision>
    <revision>
      <id>2830572</id>
      <parentid>2830568</parentid>
      <timestamp>2026-09-02T19:59:17Z</timestamp>
      <contributor>
        <username>Mitchal Dichter</username>
        <id>2824330</id>
      </contributor>
      <comment>added empty Exponents and Solving a Linear Equation page</comment>
      <origin>2830572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7056" sha1="ob05nkyll6msp1knybc7fln3uqtqdwy" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===[[/Glossary/|Glossary]]===

===Arithmetic===

===Pre-Algebra===
*[[/Pre-Algebra Exponents/|Exponents]]

===Algebra===
*[[/Algebra Solving a Linear Equation/|Solving a Linear Equation]]
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or iteratively edit and save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[/Wishlist/|Wishlist]] page where you can add text, but not a blank page, if you must. You can also request additions to existing pages, like including irrational exponents and examples in the existing [[/Exponents/|Exponents]] page. You can also contribute specific exercises and examples, like real-world applications that could interest students, and request someone translate them into TeX.</text>
      <sha1>ob05nkyll6msp1knybc7fln3uqtqdwy</sha1>
    </revision>
    <revision>
      <id>2830578</id>
      <parentid>2830572</parentid>
      <timestamp>2026-09-02T20:19:59Z</timestamp>
      <contributor>
        <username>Mitchal Dichter</username>
        <id>2824330</id>
      </contributor>
      <minor />
      <origin>2830578</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7025" sha1="mq4g65np5tmcepqx4yes6sdnkshroqr" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===Arithmetic===

===Pre-Algebra===
*[[/Pre-Algebra Exponents/|Exponents]]

===Algebra===
*[[/Algebra Solving a Linear Equation/|Solving a Linear Equation]]
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or iteratively edit and save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[/Wishlist/|Wishlist]] page where you can add text, but not a blank page, if you must. You can also request additions to existing pages, like including irrational exponents and examples in the existing [[/Exponents/|Exponents]] page. You can also contribute specific exercises and examples, like real-world applications that could interest students, and request someone translate them into TeX.</text>
      <sha1>mq4g65np5tmcepqx4yes6sdnkshroqr</sha1>
    </revision>
    <revision>
      <id>2830582</id>
      <parentid>2830578</parentid>
      <timestamp>2026-09-02T20:25:45Z</timestamp>
      <contributor>
        <username>Mitchal Dichter</username>
        <id>2824330</id>
      </contributor>
      <comment>trying to make subpages</comment>
      <origin>2830582</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7258" sha1="ocrov4e0bnlwqeu528wzihl01egoetj" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[User:Mitchal Dichter/Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===Arithmetic===

===Pre-Algebra===
*[[User:Mitchal Dichter/Pre-University Mathematics/Exponents/|Exponents]]

===Algebra===
*[[User:Mitchal Dichter/Pre-University Mathematics/Solving a Linear Equation/|Solving a Linear Equation]]
*[[User:Mitchal Dichter/Pre-University Mathematics/Algebra Functions|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or iteratively edit and save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[User:Mitchal Dichter/Pre-University Mathematics/Wishlist|Wishlist]] page where you can add text, but not a blank page, if you must. You can also request additions to existing pages, like including irrational exponents and examples in the existing [[User:Mitchal Dichter/Pre-University Mathematics/Exponents|Exponents]] page. You can also contribute specific exercises and examples, like real-world applications that could interest students, and request someone translate them into TeX.</text>
      <sha1>ocrov4e0bnlwqeu528wzihl01egoetj</sha1>
    </revision>
    <revision>
      <id>2830585</id>
      <parentid>2830582</parentid>
      <timestamp>2026-09-02T20:28:06Z</timestamp>
      <contributor>
        <username>Mitchal Dichter</username>
        <id>2824330</id>
      </contributor>
      <minor />
      <origin>2830585</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7260" sha1="aua0gswu2qlzzdlofzzjlfpuokoc0jh" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[User:Mitchal Dichter/Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===Arithmetic===

===Pre-Algebra===
*[[User:Mitchal Dichter/Pre-University Mathematics/Exponents/|Exponents]]

===Algebra===
*[[User:Mitchal Dichter/Pre-University Mathematics/Solving a Linear Equation/|Solving a Linear Equation]]
*[[User:Mitchal Dichter/Pre-University Mathematics/Algebra Functions|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or iteratively edit and save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[User:Mitchal Dichter/Pre-University Mathematics/Wishlist/|Wishlist]] page where you can add text, but not a blank page, if you must. You can also request additions to existing pages, like including irrational exponents and examples in the existing [[User:Mitchal Dichter/Pre-University Mathematics/Exponents/|Exponents]] page. You can also contribute specific exercises and examples, like real-world applications that could interest students, and request someone translate them into TeX.</text>
      <sha1>aua0gswu2qlzzdlofzzjlfpuokoc0jh</sha1>
    </revision>
    <revision>
      <id>2830589</id>
      <parentid>2830585</parentid>
      <timestamp>2026-09-02T20:29:50Z</timestamp>
      <contributor>
        <username>Mitchal Dichter</username>
        <id>2824330</id>
      </contributor>
      <minor />
      <origin>2830589</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7425" sha1="940x1yumt8jq7ac5kir3xfc9gnfl3jp" xml:space="preserve">{{mathematics}}

The goal of this Wikiversity course is to provide educational materials for pre-university mathematics. What is considered pre-university mathematics is not the same in every country, so some topics may be missing and some topics may be considered university level mathematics depending on where you live. The reader will have to use their best judgement on what content is relevant to their studies.

Please see [[User:Mitchal Dichter/Pre-University Mathematics#Contributing|Contributing]] below for compatible content rules and creating accessible content.

===Arithmetic===

===Pre-Algebra===
*[[User:Mitchal Dichter/Pre-University Mathematics/Exponents/|Exponents]]

===Algebra===
*[[User:Mitchal Dichter/Pre-University Mathematics/Solving a Linear Equation/|Solving a Linear Equation]]
*[[User:Mitchal Dichter/Pre-University Mathematics/Algebra Functions|Functions]]

===Geometry===

===Statistics===

== Contributing ==

As with almost everything on Wikiversity, contributions to this course will be licensed by both the [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0 License] and the [https://www.gnu.org/copyleft/fdl.html GFDL]. A common exception is media stored on [https://commons.wikimedia.org Wikimedia Commons] and displayed on Wikiversity, which accepts more licenses. See [https://commons.wikimedia.org/wiki/Commons:Licensing#Well-known_licenses Commons:Licensing] for more details.

Contributing text that is considered your own original work is not too difficult in mathematics, but creating original graphics is both more difficult and time consuming. Please ensure all contributions are compatible if you choose to contribute.

=== Write for English learners ===

The text is intentionally simple both to make language translation easier and to be more easily understood by readers not fluent in English. Please use short sentences, [https://en.wikipedia.org/wiki/Active_voice active voice], and simple vocabulary. Please avoid [https://en.wikipedia.org/wiki/Idiom idioms], [https://en.wikipedia.org/wiki/Slang slang], [https://en.wikipedia.org/wiki/Passive_voice passive voice], long sentences, complex sentences, 
[https://en.wikipedia.org/wiki/English_phrasal_verbs phrasal verbs], culture-specific references, ambiguous [https://en.wikipedia.org/wiki/Pronoun pronouns], and [https://en.wikipedia.org/wiki/Double_negative double negatives]. Use your best judgement on what is considered complex English based on the difficulty of the maths topic, such as simpler for a page on addition and more complex for a page on trigonometry.

=== Write for an International Audience ===

Examples
* SI units instead of the imperial system.
* Aluminium instead of aluminum.
* Yellow instead of amber.
* Maths instead of math.
* Exercises instead of problems.

=== Write for the Student but also for the Educator ===

Educational text often has an underlying pattern that guides the student but is not explicitly written. An example would be students learning how to solve a linear equation in one variable. 
* Students first use the idea of a two-pan balance scale with identical but unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, finding the weight of an &lt;math&gt;x&lt;/math&gt; by adding and removing weights from the scale so that the scale remains balanced. (Provides an accessible analogy to students before transitioning to equations.)
* Students redo the same two-pan balance scale exercises in the form of equations.
* The first set of exercises require only addition and subtraction, will have small and positive integers in every step to solve the equation, and the solution is a positive integer. (These problems are intentionally simple and students may try to guess the solution.)
* Multiplication and division are required to solve the next set of exercises and the solution is still a positive integer.
* The next set of exercises will have positive fractions as the solution, fractions when solving, or both. (Students can no longer reliably guess the solution.)
* The next set of exercises will have positive and negative integers and fractions as the solution. (Helpful analogies, like an equation as a two-pan balance scale with unknown weights &lt;math&gt;x&lt;/math&gt; and known weights, do not work well when the unknown weights &lt;math&gt;x&lt;/math&gt; have negative mass and the solution process become more abstract.)
* Equations that have zero solutions and infinite solutions are introduced. (A linear equation can have &lt;math&gt;0&lt;/math&gt;, &lt;math&gt;1&lt;/math&gt;, or infinite solutions. In the case of &lt;math&gt;0&lt;/math&gt; solutions, the equation simplifies to something like &lt;math&gt;4 = 9&lt;/math&gt;, which is not possible and no value of &lt;math&gt;x&lt;/math&gt; will solve the equation. In the case of infinite solutions, the equation starts out with identical left and right hand sides, such as &lt;math&gt;4x-3 = 4x-3&lt;/math&gt;, which will work no mater what &lt;math&gt;x&lt;/math&gt; is.)

The text in parentheses is for the educator and explicitly states why the exercises are in the order they are. The educator could be a classroom teacher or a family member of the student. The classroom educator will likely understand the reasoning of the exercises and their ordering without the text in parentheses. A family member may have learned to solve a linear equation many years ago or never learned. The text in parentheses is extremely helpful to this less qualified educator.

Please include notes to the educator like in the above example. The notes explicitly state the design decisions, how understanding is evolving in the mind of the student, and any other information that the educator can use to better teach the student.

=== Write Mathematical Symbols and Equations in TeX ===

TeX renders maths symbols in a much nicer format than regular text using &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; in the page source. Wikiversity has a page on [[Help:Formula|writing in TeX]] with many examples.

The source,

&lt;code&gt;&lt;nowiki&gt;
&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
&lt;/nowiki&gt;&lt;/code&gt;

is rendered as,

&lt;div class=&quot;center&quot;&gt;
&lt;math&gt;x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&lt;/math&gt;
&lt;/div&gt;
 
The enclosing &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;center&quot;&gt; ... &lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt; is for centering the rendered equation. The &lt;code&gt;&lt;nowiki&gt;&lt;math&gt; ... &lt;/math&gt;&lt;/nowiki&gt;&lt;/code&gt; encloses the TeX markup language. 

=== Creating New Pages ===

If you are so generous as to add a page for a fully explained topic with exercises, and real-world examples if possible, then please use the drafting features of your Wikiversity profile, or iteratively edit and save the draft locally, before posting the page.

Please do not add blank pages and expect someone else to finish them. Having incomplete pages makes the entire course look incomplete. There is a [[User:Mitchal Dichter/Pre-University Mathematics/Wishlist/|Wishlist]] page where you can add text, but not a blank page, if you must. You can also request additions to existing pages, like including irrational exponents and examples in the existing [[User:Mitchal Dichter/Pre-University Mathematics/Exponents/|Exponents]] page. (This is just an example because the [[User:Mitchal Dichter/Pre-University Mathematics/Exponents/|Exponents]] page already has a section on irrational exponenents.) You can also contribute specific exercises and examples, like real-world applications that could interest students, and request someone translate them into TeX.</text>
      <sha1>940x1yumt8jq7ac5kir3xfc9gnfl3jp</sha1>
    </revision>
  </page>
  <page>
    <title>OpenStax Nutrition for Nurses</title>
    <ns>0</ns>
    <id>330717</id>
    <revision>
      <id>2830525</id>
      <parentid>2819254</parentid>
      <timestamp>2026-09-02T14:45:49Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2830525</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1513" sha1="3j482xmga1eamzaypw95ujj1yvvjmp6" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; Nutrition for Nurses ==

== Summary ==
''Nutrition for Nurses'' is structured to support the flexible integration of nutrition content across both system-based and nursing competency-based curricula. It can be used whether nutrition is taught as a standalone course or part of another nursing course. The table of contents for ''Nutrition for Nurses'' presents content in 20 chapters, organized into 9 thematic units.

The text emphasizes evidence-based practice and holistic assessment to facilitate the integration of nutritional awareness for pre-licensure nursing students in the provision of client-centered care. ''Nutrition for Nurses'' helps students develop sound clinical judgment as well as a deep understanding of the impact of nutrition on body systems across the lifespan.

Written and reviewed by highly experienced faculty, ''Nutrition for Nurses'' includes a detailed narrative, extensive features and learning resources, and ample student support. The presentation utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN).

* [https://openstax.org/details/books/nutrition OpenStax Nutrition for Nurses] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/nutrition-for-nurses/ OpenStax Nutrition for Nurses audiobook]. Available as audio textbook.

[[Category:Nutrition]]
[[Category:OpenStax]]</text>
      <sha1>3j482xmga1eamzaypw95ujj1yvvjmp6</sha1>
    </revision>
  </page>
  <page>
    <title>OpenStax Clinical Nursing Skills</title>
    <ns>0</ns>
    <id>330718</id>
    <revision>
      <id>2830506</id>
      <parentid>2819253</parentid>
      <timestamp>2026-09-02T14:21:27Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2830506</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1882" sha1="6t77v3qqisp68c2yqw9r7l9peojvbes" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Clinical Nursing Skills&lt;/big&gt; ==

== Summary ==
''Clinical Nursing Skills'' is designed to equip nursing students with the practical knowledge and hands-on skills necessary to provide comprehensive patient care. The material emphasizes the application of clinical judgment in a variety of settings, ensuring that students are prepared to deliver high-quality care across different patient populations and clinical scenarios. The content utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN).
''Clinical Nursing Skills'' provides detailed instructions on basic procedures such as hygiene, mobility, vital signs assessment, medication administration, and wound care. It also guides students through more complex skills, including intravenous therapy, catheterization, tracheostomy care, and emergency interventions. By integrating the Clinical Judgment Measurement Model, the material helps students recognize, analyze, prioritize, create, act, and evaluate outcomes in various clinical situations, fostering critical thinking and clinical decision making. By studying ''Clinical Nursing Skills'', students will gain the confidence and competence needed to perform essential nursing tasks, make informed clinical decisions, and provide compassionate, patient-centered care, which will prepare students for success in their clinical rotations and future professional practice.

* [https://openstax.org/details/books/clinical-nursing-skills OpenStax ''Clinical Nursing Skills''] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/clinical-nursing-skills/ OpenStax ''Clinical Nursing Skills'' audiobook] Available as audio textbook.

[[Category:Nursing]]
[[Category:OpenStax]]</text>
      <sha1>6t77v3qqisp68c2yqw9r7l9peojvbes</sha1>
    </revision>
  </page>
  <page>
    <title>User:U3285438</title>
    <ns>2</ns>
    <id>330795</id>
    <revision>
      <id>2830679</id>
      <parentid>2830416</parentid>
      <timestamp>2026-09-03T05:37:31Z</timestamp>
      <contributor>
        <username>U3285438</username>
        <id>3103750</id>
      </contributor>
      <comment>/* Social contributions */ +1</comment>
      <origin>2830679</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3663" sha1="qugy7o8wfza0vbzwukuwk70cg37a0t2" xml:space="preserve">== About me ==
Hello! My name is Zara (''she/her''). 

I am a '''third-year student''' at the [https://www.canberra.edu.au/ University of Canberra (UC)], studying a Bachelor of Science in Psychology with a Breadth Major in Counselling Studies. I am hoping to undertake postgraduate studies in [[wikipedia:Clinical_psychology|clinical psychology]], and I am particularly '''interested in working with children and adolescents'''. 
[[File:Canadian swimmer at 2019 international tournament.jpg|thumb|150x150px|'''Figure 1.''' As a competitive swimmer, my favourite stroke was butterfly.  ]]
I am currently undertaking the unit [[Motivation and emotion|Motivation and Emotion]] at UC. 

Some of my hobbies include:  

* Reading
* [[wikipedia:Swimming|Swimming]] (see Figure 1)

== Book chapter I'm working on ==
I am writing a book chapter on [[Motivation and emotion/Book/2026/Emotion dysregulation|emotion dysregulation]] for the [[Motivation and emotion/Book|Motivation and emotion book]] project. Through this chapter I will be exploring the question &quot;what is emotion dysregulation, what are its consequences and how can it be managed?&quot;  

My interest in this topic stems from my experience as a classroom support assistant in a primary school. Within this role, I work with young students to help build emotional literacy and support them in regulating their emotions. I am eager to further my understanding of emotion dysregulation and share this knowledge with others.   

== Social contributions ==

#[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261?entry_id=805511 Posted in a UCLearn discussion thread to share the areas of motivation and emotion I am interested in exploring this semester.]  (18 Aug. 2026)
#[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261?entry_id=805526 Responded to a peer's discussion post on UCLearn.] (18 Aug. 2026)
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2823381&amp;oldid=2762016 Fixed grammatical error to improve clarity.] (20 Aug. 2026)
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FTo-do_lists&amp;diff=2823387&amp;oldid=2672363 Made small stylistic correction in formatting.] (20 Aug. 2026)
#[https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2025%2FPain_avoidance_motivation&amp;diff=2823539&amp;oldid=2758910 Recommended a relevant book chapter for inclusion in the 'See also' section.] (20 Aug. 2026)
#[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCoercive_control_in_intimate_partner_violence&amp;diff=2825669&amp;oldid=2811025 Edited a past book chapter's 'See also' section to include a missing source citation in parentheses after the relevant link.] (25 Aug. 2026)
#[https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FBuilding_therapeutic_alliance&amp;diff=2830378&amp;oldid=2829992 Offered a relevant therapeutic framework that could be included in my peer's book chapter.] (02 Sept. 2026)
#[[Talk:Motivation and emotion/Book/2026/Fear extinction#c-U3285438-20260902052100-Food for thought: Integrating Pavlov's classical conditioning|Suggested integrating a relevant theory to help contextualise key concepts in my peer's book chapter.]] (02 Sept. 2026)
#[[Talk:Motivation and emotion/Book/2026/Sun exposure and protection motivation#c-U3285438-20260903051900-Food for thought: The health belief model|Suggested including a relevant theory (the health belief model) in my peer's book chapter as an additional lens for understanding health behaviour change]] (03 Sept. 2026)</text>
      <sha1>qugy7o8wfza0vbzwukuwk70cg37a0t2</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Positive emotion dysregulation</title>
    <ns>0</ns>
    <id>331035</id>
    <revision>
      <id>2830635</id>
      <parentid>2830196</parentid>
      <timestamp>2026-09-03T02:27:12Z</timestamp>
      <contributor>
        <username>P U3270518</username>
        <id>3106535</id>
      </contributor>
      <comment>Copilot was used to get overview on how i can approach this section and was also used to make paragraph shorter. Few details about my scenorio was provided to copilot to understand how i can make it more precise. Link to the conversation: https://copilot.cloud.microsoft/chat/conversation/b2c99845-c7f8-4821-ae0c-548d3a440bc0?fromcode=cmmqbg1y3ah&amp;es=UnauthClick&amp;login_hint=u3270518%40uni.canberra.edu.au</comment>
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      <text bytes="14658" sha1="nrlec605xvomu0ba3id587zg4v04ix3" xml:space="preserve">{{title|Positive emotion dysregulation:&lt;br&gt;What is positive emotion dysregulation and how does it affect psychological functioning?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=8}}
[[File:Woman in red sweater with hand in air.jpg|right|thumb|250px|'''Figure 1.''' Grace experiencing happiness and excitement following several positive life events.]]

'''Scenario'''

Grace is a university student who feels that she has just won the lottery of her life. First, she receives a PhD offer from her dream university. Shortly afterwards, she learns that she has been awarded a scholarship to support her studies. 

For next several days, Grace feels unusually happy and energised (see Figure 1). She sleeps very little because she believes there is too much to achieve. To celebrate her success, she purchases an expensive laptop, makes an impulsive travel booking and commits to several university projects despite having a full schedule. 

People close to Grace begin to wonder if something is not right. Her close friends tried to slow her down, but she ignored them. From Grace's perspective, she is just happy, motivated and confident as pieces of her life are finally coming together in a positive way. 

💡 What is happening to Grace?

💡 Why are her positive emotions become so intense?

{{RoundBoxBottom}}

[[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] are a normal and important part of human life. It includes a variety of feelings such as joy, interest, love, contentment, gratitude, awe, and amusement (Campos et al., 2013). These emotions play an important role in expanding people's thoughts and possible actions to help them build lasting psychological and social resources (Fredickson, 2001). However, positive emotions might not always be beneficial if they are inadequately regulated in terms of magnitude, context, and duration. A review by Gruber et al. (2020) on positive emotion disturbance suggests that extremely strong or prolonged positive emotional states can sometimes interfere with adaptive functioning. Positive emotion dysregulation helps to explain how positive emotional states can feel initially feel good but can sometimes contribute to harmful thoughts and behaviour. This chapter will explore how positive emotions can shift from being adaptive to maladaptive, and how it can affect psychological functioning.  

{{RoundBoxTop|theme=2}}
🔎 '''Focus questions'''

* What is positive emotion dysregulation, and how it can be distinguished from healthy positive emotions?
* What psychological and biological factors contribute to positive emotion dysregulation?
* How does positive emotion dysregulation manifest in thoughts, motivation, and behaviour?
* How can positive emotion dysregulation affect psychological functioning and wellbeing? 

{{RoundBoxBottom}}

==Positive emotion: From benefit to dysregulation==
Understanding positive emotion dysregulation requires more than just identifying whether an emotional experience is pleasant or intense. Positive emotions are usually considered pleasant in nature. But impact of positive emotions will be always relative to how it is integrated into cognitive, behavioural, goal-directed and other factors. This section will consider why positive emotions are important, how they are normally regulated and how difficulties in regulating them can lead to positive emotion dysregulation.  

=== Positive emotions and its adaptive functions ===

* Positive emotions play an important role in psychological functioning. They include numerous experiences such as joy, interest, love, contentment, gratitude, awe, and amusement which can influence cognition, behaviour, and social interaction (Campos et al., 2013). Instead of simply producing pleasant effects, positive emotions can encourage individuals to explore their environment, learn from experiences, engage with others and pursue meaningful goals.
* Positive emotions can broaden people's thoughts, cognition and behaviour. [[wikipedia:Broaden-and-build|Broaden-and-Build Theory]] by Fredrickson (2001) proposes that positive emotions broaden momentary thought-action repertoires that encourages exploration, flexibility and engagement.
* Positive emotions can help to build psychological and social resources, including relationships and coping resources (Fredrickson, 2001).
* Positive emotions are not always beneficial in every circumstances. Effects of positive emotions may depend on intensity, timing, and context. This provides a foundation for understanding when positive emotion may become less adaptive (Gruber et al., 2011).

=== Understanding emotion regulation ===

* Emotional regulation is one’s ability to seek control over own emotional state (Gross, 2015). The process model of emotion regulation by Gross (1998) provides a framework for understanding how individual can modify emotional experiences.
* Positive emotions also require regulation. According to Carl et al., (2013), individuals may need to regulate their positive emotions based on their goals and circumstances. 
* Adaptive emotion regulation is more flexible in nature compared to emotion reduction. Being able to regulate one’s positive emotions can be helpful in ensuring effective functioning.

=== From positive emotions to dysregulation ===

* An intense positive emotion does not necessarily means dysregulated. Feelings such as happiness, excitement, or enthusiasm may be well-placed responses to certain life events.
* Context, intensity, persistence and implications are crucial. Positive emotions can be considered maladaptive if are poorly fitted to the situation or if they start to interfere with functioning (Gruber et al., 2011). 
* The topic of positive emotion dysregulation is a relatively new area of interest. A review by Vogue et al. (2023), suggests that there is a need to examine the challenges of regulating positive emotions rather than focusing exclusively on negative emotions. 

== Mechanisms underlying positive emotion dysregulation ==
 
Positive emotion dysregulation is unlikely result from a single factor. Instead, it is a result of psychological, physiological, biological, individual, and environment factors. All these factors may interact to influence how strongly positive emotions are experienced and how they are regulated.  

{{ic|An alternative approach here could be to focus on the main theory or theories used to understand PED. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt;}}

(Not sure if section is needed to answer the topic question). Might keep it or remove it for book chapter for more clarity. 

Consfused between these two format: 

Format 1: (Focus question 1: What is it?                                               

Focus question 2: How does it manifest?                              

Focus question 3: How does it affect functioning?)                                

Format 2: (Focus question 1: What is it? 

Focus question 2: Why it is happening? 

Focus question 3: How does it manifest? 

Focus question 4: How does it affect functioning?) 

=== Psychological factors ===

* Emotion-regulation processes may influence how positive emotional states are managed. 
* Reward sensitivity may influence responses to positive experiences.
* Motivational and cognitive processes may shape the results of positive emotion.

=== Physiological and biological factors ===

* Positive emotions include measurable physiological and neural responses.
* Biological reward system may contribute to positive emotional responding.
* Little research available in the field of positive emotion dysregulation.

=== Individual and environmental factors ===

* Temperament may influence positive emotional reactivity (Vogel et al., 2023).
* Individual and contextual factors influences emotion regulation. 
* Situation can determine whether positive emotional activation is adaptive. 

== Consequences of positive emotion dysregulation ==

The regulation of positive emotions may manifest itself through altered thinking processes, decision-making, goals pursued and behaviour of an individual. While positive emotions helps to expand attention and motivate individuals to explore more, strong and poorly regulated emotional activation may influence judgement. This section will discuss the way in which positive experiences may impact cognition, motivation and decision-making processes.

=== Changes in cognition and decision-making ===

* Positive emotions can broaden attention and action thought repertoires. Fredrickson and Branigan (2005) found that positive affective states broadened both attention and the range of thought-action responses that the participants gave. It provided empirical support for broaden-and-build theory. 
* Positive emotion dysregulation can influence decision-making.
* The effects of cognitive broadening are context-dependent 

=== Goal pursuit and behavioural activation ===

* The positive emotions would enhance approach motivation and behavioural activation. 
* Excessive activation might help achieve goals but could be exaggerated. In case of exaggeration, the activation might make people over-commit.
* Grace’s behaviour explains this as initially she was motivated to participate in university opportunities. But repeatedly taking additional projects despite limited time shows that goal directed activation is becoming difficult to regulate. 

=== Risk-taking and Impulsivity ===

* Presence of positive emotional activation may lead to increased level of impulsivity and risky behaviour. 
* Positive urgency provides a precise reason for rash behaviour during high levels of positive emotions. Cyders and Smith (2008) define positive urgency as the tendency of behaving rashly during high levels of positive affect. This behavioural trend can be risky as well. 
* Impulsivity and risk-taking should not be equated with positive emotion dysregulation  

== Positive emotion dysregulation and psychological functioning ==
{{expand}}

=== Positive emotions as good psychological resources ===

* Positive emotions can support engagement and goal-directed functioning.
* Valuable for academic achievement.
* Encourage exploration and persistence.

=== When positive emotional experiences becomes difficult to regulate ===

* Poorly regulated positive emotions can create interpersonal difficulties.
* Positive emotion dysregulation may interfere with everyday responsibilities.
* It can affect sleep and concentration 

;Quiz
Choose your answer and click &quot;Submit&quot;
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement explains why positive emotion dysregulation can affect psychological functioning?
|type=&quot;()&quot;}
+ Poorly regulated positive emotion may interfere with wellbeing, relationships and responsibilities
- Positive emotions are always harmful
- Positive emotions have no effect on behaviour
- Intense positive emotions always indicate dysregulation

&lt;/quiz&gt;

==Conclusion==
*Positive emotion dysregulation is not simply experiencing intense happiness, but difficulty in regulating positive emotions.
*Dysregulation can influence thoughts, decision-making and behaviour.
*Difficulty in regulating positive emotions may affect everyday responsibilities and functioning. 

==See also ==
*[[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory of positive emotions]] (Book chapter, 2016)
*[[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion Dysregulation]] (Book chapter, 2026)
*[[wikipedia:Emotional_dysregulation|Emotional Dysregulation]] (Wikipedia)
*[[Motivation and emotion/Book/2020/Positive emotion|Positive emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin'', ''134''(6), 807–828. https://doi.org/10.1037/a0013341

Campos, B., Shiota, M. N., Keltner, D., Gonzaga, G. C., &amp; Goetz, J. L. (2013). What is shared, what is different? Core relational themes and expressive displays of eight positive emotions. ''Cognition and Emotion'', ''27''(1), 37–52. https://doi.org/10.1080/02699931.2012.683852

Carl, J. R., Soskin, D. P., Kerns, C., &amp; Barlow, D. H. (2013). Positive emotion regulation in emotional disorders: A theoretical review. ''Clinical Psychology Review'', ''33''(3), 343–360. https://doi.org/10.1016/j.cpr.2013.01.003

Fredrickson B. L. (2001). The role of positive emotions in positive psychology. The broaden-and-build theory of positive emotions. ''The American psychologist'', ''56''(3), 218–226. https://doi.org/10.1037//0003-066x.56.3.218

Fredrickson, B. L., &amp; Branigan, C. (2005). Positive emotions broaden the scope of attention and thought‐action repertoires. ''Cognition &amp; Emotion'', ''19''(3), 313–332. https://doi.org/10.1080/02699930441000238

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271

Gruber, J., Mauss, I. B., &amp; Tamir, M. (2011). A Dark Side of Happiness? How, When, and Why Happiness Is Not Always Good. ''Perspectives on Psychological Science'', ''6''(3), 222–233. https://doi.org/10.1177/1745691611406927

Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry'', ''26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Gruber, J., Villanueva, C., Burr, E., Purcell, J. R., &amp; Karoly, H. (2020). Understanding and Taking Stock of Positive Emotion Disturbance. ''Social and personality psychology compass'', ''14''(1), e12515. https://doi.org/10.1111/spc3.12515

Vogel, A. C., Brotman, M. A., Roy, A. K., &amp; Perlman, S. B. (2023). Review: Defining positive emotion dysregulation: Integrating temperamental and clinical perspectives. ''Journal of the American Academy of Child &amp; Adolescent Psychiatry'', ''62''(3), 297–305. https://doi.org/10.1016/j.jaac.2022.06.019

}}

==External links==

* [https://www.psychologytoday.com/au/blog/everyday-resilience/202404/emotional-well-being-5-healthy-practices-for-regulation Emotional Wellbeing] (Psychology Today)
* [https://www.youtube.com/watch?v=MyfzIQH6YKI Positive Emotions with Barbara Fredrickson] (Youtube.com)
* [https://sk.sagepub.com/ency/edvol/the-sage-encyclopedia-of-lifespan-human-development/chpt/reward-sensitivity Reward Sensitivity] (sk.sagepub.com)

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      <text bytes="14709" sha1="ox8dbm1bgcru66tnsk5jukhtc8kvka8" xml:space="preserve">{{title|Positive emotion dysregulation:&lt;br&gt;What is positive emotion dysregulation and how does it affect psychological functioning?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=8}}
[[File:Woman in red sweater with hand in air.jpg|right|thumb|250px|'''Figure 1.''' Grace experiencing happiness and excitement following several positive life events.]]

'''Scenario'''

Grace is a university student who feels that she has just won the lottery of her life. First, she receives a PhD offer from her dream university. Shortly afterwards, she learns that she has been awarded a scholarship to support her studies. 

For next several days, Grace feels unusually happy and energised (see Figure 1). She sleeps very little because she believes there is too much to achieve. To celebrate her success, she purchases an expensive laptop, makes an impulsive travel booking and commits to several university projects despite having a full schedule. 

People close to Grace begin to wonder if something is not right. Her close friends tried to slow her down, but she ignored them. From Grace's perspective, she is just happy, motivated and confident as pieces of her life are finally coming together in a positive way. 

💡 What is happening to Grace?

💡 Why are her positive emotions become so intense?

{{RoundBoxBottom}}

[[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] are a normal and important part of human life. It includes a variety of feelings such as joy, interest, love, contentment, gratitude, awe, and amusement (Campos et al., 2013). These emotions play an important role in expanding people's thoughts and possible actions to help them build lasting psychological and social resources (Fredickson, 2001). However, positive emotions might not always be beneficial if they are inadequately regulated in terms of magnitude, context, and duration. A review by Gruber et al. (2020) on positive emotion disturbance suggests that extremely strong or prolonged positive emotional states can sometimes interfere with adaptive functioning. Positive emotion dysregulation helps to explain how positive emotional states can feel initially feel good but can sometimes contribute to harmful thoughts and behaviour. This chapter will explore how positive emotions can shift from being adaptive to maladaptive, and how it can affect psychological functioning.  

{{RoundBoxTop|theme=2}}
🔎 '''Focus questions'''

* What is positive emotion dysregulation, and how it can be distinguished from healthy positive emotions?
* What psychological and biological factors contribute to positive emotion dysregulation?
* How does positive emotion dysregulation manifest in thoughts, motivation, and behaviour?
* How can positive emotion dysregulation affect psychological functioning and wellbeing? 

{{RoundBoxBottom}}

==Positive emotion: From benefit to dysregulation==
Understanding positive emotion dysregulation requires more than just identifying whether an emotional experience is pleasant or intense. Positive emotions are usually considered pleasant in nature. But impact of positive emotions will be always relative to how it is integrated into cognitive, behavioural, goal-directed and other factors. This section will consider why positive emotions are important, how they are normally regulated and how difficulties in regulating them can lead to positive emotion dysregulation.  

=== Positive emotions and its adaptive functions ===

* Positive emotions play an important role in psychological functioning. They include numerous experiences such as joy, interest, love, contentment, gratitude, awe, and amusement which can influence cognition, behaviour, and social interaction (Campos et al., 2013). Instead of simply producing pleasant effects, positive emotions can encourage individuals to explore their environment, learn from experiences, engage with others and pursue meaningful goals.
* Positive emotions can broaden people's thoughts, cognition and behaviour. [[wikipedia:Broaden-and-build|Broaden-and-Build Theory]] by Fredrickson (2001) proposes that positive emotions broaden momentary thought-action repertoires that encourages exploration, flexibility and engagement.
* Positive emotions can help to build psychological and social resources, including relationships and coping resources (Fredrickson, 2001).
* Positive emotions are not always beneficial in every circumstances. Effects of positive emotions may depend on intensity, timing, and context. This provides a foundation for understanding when positive emotion may become less adaptive (Gruber et al., 2011).

=== Understanding emotion regulation ===

* Emotional regulation is one’s ability to seek control over own emotional state (Gross, 2015). The process model of emotion regulation by Gross (1998) provides a framework for understanding how individual can modify emotional experiences.
* Positive emotions also require regulation. According to Carl et al., (2013), individuals may need to regulate their positive emotions based on their goals and circumstances. 
* Adaptive emotion regulation is more flexible in nature compared to emotion reduction. Being able to regulate one’s positive emotions can be helpful in ensuring effective functioning.

=== From positive emotions to dysregulation ===

* An intense positive emotion does not necessarily means dysregulated. Feelings such as happiness, excitement, or enthusiasm may be well-placed responses to certain life events.
* Context, intensity, persistence and implications are crucial. Positive emotions can be considered maladaptive if are poorly fitted to the situation or if they start to interfere with functioning (Gruber et al., 2011). 
* The topic of positive emotion dysregulation is a relatively new area of interest. A review by Vogue et al. (2023), suggests that there is a need to examine the challenges of regulating positive emotions rather than focusing exclusively on negative emotions. 

== Mechanisms underlying positive emotion dysregulation ==
 
Positive emotion dysregulation is unlikely result from a single factor. Instead, it is a result of psychological, physiological, biological, individual, and environment factors. All these factors may interact to influence how strongly positive emotions are experienced and how they are regulated.  

{{ic|An alternative approach here could be to focus on the main theory or theories used to understand PED. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt;}}

(Not sure if section is needed to answer the topic question). Might keep it or remove it for book chapter for more clarity. 

Consfused between these two format: 

Format 1: (Focus question 1: What is it?                                               

Focus question 2: How does it manifest?                              

Focus question 3: How does it affect functioning?)                                

Format 2: (Focus question 1: What is it? 

Focus question 2: Why it is happening? 

Focus question 3: How does it manifest? 

Focus question 4: How does it affect functioning?) 

=== Psychological factors ===

* Emotion-regulation processes may influence how positive emotional states are managed. 
* Reward sensitivity may influence responses to positive experiences.
* Motivational and cognitive processes may shape the results of positive emotion.

=== Physiological and biological factors ===

* Positive emotions include measurable physiological and neural responses.
* Biological reward system may contribute to positive emotional responding.
* Little research available in the field of positive emotion dysregulation.

=== Individual and environmental factors ===

* Temperament may influence positive emotional reactivity (Vogel et al., 2023).
* Individual and contextual factors influences emotion regulation. 
* Situation can determine whether positive emotional activation is adaptive. 

== Consequences of positive emotion dysregulation ==

The regulation of positive emotions may manifest itself through altered thinking processes, decision-making, goals pursued and behaviour of an individual. While positive emotions helps to expand attention and motivate individuals to explore more, strong and poorly regulated emotional activation may influence judgement. This section will discuss the way in which positive experiences may impact cognition, motivation and decision-making processes.

=== Changes in cognition and decision-making ===

* Positive emotions can broaden attention and action thought repertoires. Fredrickson and Branigan (2005) found that positive affective states broadened both attention and the range of thought-action responses that the participants gave. It provided empirical support for broaden-and-build theory. 
* Positive emotion dysregulation can influence decision-making.
* The effects of cognitive broadening are context-dependent 

=== Goal pursuit and behavioural activation ===

* The positive emotions would enhance approach motivation and behavioural activation. 
* Excessive activation might help achieve goals but could be exaggerated. In case of exaggeration, the activation might make people over-commit.
* Grace’s behaviour explains this as initially she was motivated to participate in university opportunities. But repeatedly taking additional projects despite limited time shows that goal directed activation is becoming difficult to regulate. 

=== Risk-taking and Impulsivity ===

* Presence of positive emotional activation may lead to increased level of impulsivity and risky behaviour. 
* Positive urgency provides a precise reason for rash behaviour during high levels of positive emotions. Cyders and Smith (2008) define positive urgency as the tendency of behaving rashly during high levels of positive affect. This behavioural trend can be risky as well. 
* Impulsivity and risk-taking should not be equated with positive emotion dysregulation  

== Positive emotion dysregulation and psychological functioning ==
{{expand}}

=== Positive emotions as good psychological resources ===

* Positive emotions can support engagement and goal-directed functioning.
* Valuable for academic achievement.
* Encourage exploration and persistence.

=== When positive emotional experiences becomes difficult to regulate ===

* Poorly regulated positive emotions can create interpersonal difficulties.
* Positive emotion dysregulation may interfere with everyday responsibilities.
* It can affect sleep and concentration 

;Quiz
Choose your answer and click &quot;Submit&quot;
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement explains why positive emotion dysregulation can affect psychological functioning?
|type=&quot;()&quot;}
+ Poorly regulated positive emotion may interfere with wellbeing, relationships and responsibilities
- Positive emotions are always harmful
- Positive emotions have no effect on behaviour
- Intense positive emotions always indicate dysregulation

&lt;/quiz&gt;

==Conclusion==
*Positive emotion dysregulation is not simply experiencing intense happiness, but difficulty in regulating positive emotions.
*Dysregulation can influence thoughts, decision-making and behaviour.
*Difficulty in regulating positive emotions may affect everyday responsibilities and functioning. 

==See also ==
*[[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory of positive emotions]] (Book chapter, 2016)
*[[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion Dysregulation]] (Book chapter, 2026)
*[[wikipedia:Emotional_dysregulation|Emotional Dysregulation]] (Wikipedia)
*[[Motivation and emotion/Book/2020/Positive emotion|Positive emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin'', ''134''(6), 807–828. https://doi.org/10.1037/a0013341

Campos, B., Shiota, M. N., Keltner, D., Gonzaga, G. C., &amp; Goetz, J. L. (2013). What is shared, what is different? Core relational themes and expressive displays of eight positive emotions. ''Cognition and Emotion'', ''27''(1), 37–52. https://doi.org/10.1080/02699931.2012.683852

Carl, J. R., Soskin, D. P., Kerns, C., &amp; Barlow, D. H. (2013). Positive emotion regulation in emotional disorders: A theoretical review. ''Clinical Psychology Review'', ''33''(3), 343–360. https://doi.org/10.1016/j.cpr.2013.01.003

Fredrickson B. L. (2001). The role of positive emotions in positive psychology. The broaden-and-build theory of positive emotions. ''The American psychologist'', ''56''(3), 218–226. https://doi.org/10.1037//0003-066x.56.3.218

Fredrickson, B. L., &amp; Branigan, C. (2005). Positive emotions broaden the scope of attention and thought‐action repertoires. ''Cognition &amp; Emotion'', ''19''(3), 313–332. https://doi.org/10.1080/02699930441000238

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271

Gruber, J., Mauss, I. B., &amp; Tamir, M. (2011). A Dark Side of Happiness? How, When, and Why Happiness Is Not Always Good. ''Perspectives on Psychological Science'', ''6''(3), 222–233. https://doi.org/10.1177/1745691611406927

Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry'', ''26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Gruber, J., Villanueva, C., Burr, E., Purcell, J. R., &amp; Karoly, H. (2020). Understanding and Taking Stock of Positive Emotion Disturbance. ''Social and personality psychology compass'', ''14''(1), e12515. https://doi.org/10.1111/spc3.12515

Vogel, A. C., Brotman, M. A., Roy, A. K., &amp; Perlman, S. B. (2023). Review: Defining positive emotion dysregulation: Integrating temperamental and clinical perspectives. ''Journal of the American Academy of Child &amp; Adolescent Psychiatry'', ''62''(3), 297–305. https://doi.org/10.1016/j.jaac.2022.06.019

}}

==External links==

* [https://www.psychologytoday.com/au/blog/everyday-resilience/202404/emotional-well-being-5-healthy-practices-for-regulation Emotional Wellbeing] (Psychology Today)
* [https://www.youtube.com/watch?v=MyfzIQH6YKI Positive Emotions with Barbara Fredrickson] (Youtube.com)
* [https://sk.sagepub.com/ency/edvol/the-sage-encyclopedia-of-lifespan-human-development/chpt/reward-sensitivity Reward Sensitivity] (sk.sagepub.com)
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__TOC__

==Overview==
{{RoundBoxTop|theme=8}}
[[File:Woman in red sweater with hand in air.jpg|right|thumb|250px|'''Figure 1.''' Grace experiencing happiness and excitement following several positive life events.]]

'''Scenario'''

Grace is a university student who feels that she has just won the lottery of her life. First, she receives a PhD offer from her dream university. Shortly afterwards, she learns that she has been awarded a scholarship to support her studies. 

For next several days, Grace feels unusually happy and energised (see Figure 1). She sleeps very little because she believes there is too much to achieve. To celebrate her success, she purchases an expensive laptop, makes an impulsive travel booking and commits to several university projects despite having a full schedule. 

People close to Grace begin to wonder if something is not right. Her close friends tried to slow her down, but she ignored them. From Grace's perspective, she is just happy, motivated and confident as pieces of her life are finally coming together in a positive way. 

💡 What is happening to Grace?

💡 Why are her positive emotions become so intense?

{{RoundBoxBottom}}

[[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] are a normal and important part of human life. It includes a variety of feelings such as joy, interest, love, contentment, gratitude, awe, and amusement (Campos et al., 2013). These emotions play an important role in expanding people's thoughts and possible actions to help them build lasting psychological and social resources (Fredickson, 2001). However, positive emotions might not always be beneficial if they are inadequately regulated in terms of magnitude, context, and duration. A review by Gruber et al. (2020) on positive emotion disturbance suggests that extremely strong or prolonged positive emotional states can sometimes interfere with adaptive functioning. Positive emotion dysregulation helps to explain how positive emotional states can feel initially feel good but can sometimes contribute to harmful thoughts and behaviour. This chapter will explore how positive emotions can shift from being adaptive to maladaptive, and how it can affect psychological functioning.  

{{RoundBoxTop|theme=2}}
🔎 '''Focus questions'''

* What is positive emotion dysregulation, and how it can be distinguished from healthy positive emotions?
* What psychological and biological factors contribute to positive emotion dysregulation?
* How does positive emotion dysregulation manifest in thoughts, motivation, and behaviour?
* How can positive emotion dysregulation affect psychological functioning and wellbeing? 

{{RoundBoxBottom}}

==Positive emotion: From benefit to dysregulation==
Understanding positive emotion dysregulation requires more than just identifying whether an emotional experience is pleasant or intense. Positive emotions are usually considered pleasant in nature. But impact of positive emotions will be always relative to how it is integrated into cognitive, behavioural, goal-directed and other factors. This section will consider why positive emotions are important, how they are normally regulated and how difficulties in regulating them can lead to positive emotion dysregulation.  

=== Positive emotions and its adaptive functions ===

* Positive emotions play an important role in psychological functioning. They include numerous experiences such as joy, interest, love, contentment, gratitude, awe, and amusement which can influence cognition, behaviour, and social interaction (Campos et al., 2013). Instead of simply producing pleasant effects, positive emotions can encourage individuals to explore their environment, learn from experiences, engage with others and pursue meaningful goals.
* Positive emotions can broaden people's thoughts, cognition and behaviour. [[wikipedia:Broaden-and-build|Broaden-and-Build Theory]] by Fredrickson (2001) proposes that positive emotions broaden momentary thought-action repertoires that encourages exploration, flexibility and engagement.
* Positive emotions can help to build psychological and social resources, including relationships and coping resources (Fredrickson, 2001).
* Positive emotions are not always beneficial in every circumstances. Effects of positive emotions may depend on intensity, timing, and context. This provides a foundation for understanding when positive emotion may become less adaptive (Gruber et al., 2011).

=== Understanding emotion regulation ===

* Emotional regulation is one’s ability to seek control over own emotional state (Gross, 2015). The process model of emotion regulation by Gross (1998) provides a framework for understanding how individual can modify emotional experiences.
* Positive emotions also require regulation. According to Carl et al., (2013), individuals may need to regulate their positive emotions based on their goals and circumstances. 
* Adaptive emotion regulation is more flexible in nature compared to emotion reduction. Being able to regulate one’s positive emotions can be helpful in ensuring effective functioning.

=== From positive emotions to dysregulation ===

* An intense positive emotion does not necessarily means dysregulated. Feelings such as happiness, excitement, or enthusiasm may be well-placed responses to certain life events.
* Context, intensity, persistence and implications are crucial. Positive emotions can be considered maladaptive if are poorly fitted to the situation or if they start to interfere with functioning (Gruber et al., 2011). 
* The topic of positive emotion dysregulation is a relatively new area of interest. A review by Vogue et al. (2023), suggests that there is a need to examine the challenges of regulating positive emotions rather than focusing exclusively on negative emotions. 

== Mechanisms underlying positive emotion dysregulation ==
 
Positive emotion dysregulation is unlikely result from a single factor. Instead, it is a result of psychological, physiological, biological, individual, and environment factors. All these factors may interact to influence how strongly positive emotions are experienced and how they are regulated.  

{{ic|An alternative approach here could be to focus on the main theory or theories used to understand PED. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt;}}

(Not sure if section is needed to answer the topic question). Might keep it or remove it for book chapter for more clarity. 

Consfused between these two format: 

Format 1: (Focus question 1: What is it?                                               

Focus question 2: How does it manifest?                              

Focus question 3: How does it affect functioning?)                                

Format 2: (Focus question 1: What is it? 

Focus question 2: Why it is happening? 

Focus question 3: How does it manifest? 

Focus question 4: How does it affect functioning?) 

=== Psychological factors ===

* Emotion-regulation processes may influence how positive emotional states are managed. 
* Reward sensitivity may influence responses to positive experiences.
* Motivational and cognitive processes may shape the results of positive emotion.

=== Physiological and biological factors ===

* Positive emotions include measurable physiological and neural responses.
* Biological reward system may contribute to positive emotional responding.
* Little research available in the field of positive emotion dysregulation.

=== Individual and environmental factors ===

* Temperament may influence positive emotional reactivity (Vogel et al., 2023).
* Individual and contextual factors influences emotion regulation. 
* Situation can determine whether positive emotional activation is adaptive. 

== Consequences of positive emotion dysregulation ==

The regulation of positive emotions may manifest itself through altered thinking processes, decision-making, goals pursued and behaviour of an individual. While positive emotions helps to expand attention and motivate individuals to explore more, strong and poorly regulated emotional activation may influence judgement. This section will discuss the way in which positive experiences may impact cognition, motivation and decision-making processes.

=== Changes in cognition and decision-making ===

* Positive emotions can broaden attention and action thought repertoires. Fredrickson and Branigan (2005) found that positive affective states broadened both attention and the range of thought-action responses that the participants gave. It provided empirical support for broaden-and-build theory. 
* Positive emotion dysregulation can influence decision-making.
* The effects of cognitive broadening are context-dependent 

=== Goal pursuit and behavioural activation ===

* The positive emotions would enhance approach motivation and behavioural activation. 
* Excessive activation might help achieve goals but could be exaggerated. In case of exaggeration, the activation might make people over-commit.
* Grace’s behaviour explains this as initially she was motivated to participate in university opportunities. But repeatedly taking additional projects despite limited time shows that goal directed activation is becoming difficult to regulate. 

=== Risk-taking and Impulsivity ===

* Presence of positive emotional activation may lead to increased level of impulsivity and risky behaviour. 
* Positive urgency provides a precise reason for rash behaviour during high levels of positive emotions. Cyders and Smith (2008) define positive urgency as the tendency of behaving rashly during high levels of positive affect. This behavioural trend can be risky as well. 
* Impulsivity and risk-taking should not be equated with positive emotion dysregulation  

== Positive emotion dysregulation and psychological functioning ==
{{expand}}

=== Positive emotions as good psychological resources ===

* Positive emotions can support engagement and goal-directed functioning.
* Valuable for academic achievement.
* Encourage exploration and persistence.

=== When positive emotional experiences becomes difficult to regulate ===

* Poorly regulated positive emotions can create interpersonal difficulties.
* Positive emotion dysregulation may interfere with everyday responsibilities.
* It can affect sleep and concentration 

;Quiz
Choose your answer and click &quot;Submit&quot;
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement explains why positive emotion dysregulation can affect psychological functioning?
|type=&quot;()&quot;}
+ Poorly regulated positive emotion may interfere with wellbeing, relationships and responsibilities
- Positive emotions are always harmful
- Positive emotions have no effect on behaviour
- Intense positive emotions always indicate dysregulation

&lt;/quiz&gt;

==Conclusion==
*Positive emotion dysregulation is not simply experiencing intense happiness, but difficulty in regulating positive emotions.
*Dysregulation can influence thoughts, decision-making and behaviour.
*Difficulty in regulating positive emotions may affect everyday responsibilities and functioning. 

==See also ==
*[[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory of positive emotions]] (Book chapter, 2016)
*[[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion Dysregulation]] (Book chapter, 2026)
*[[wikipedia:Emotional_dysregulation|Emotional Dysregulation]] (Wikipedia)
*[[Motivation and emotion/Book/2020/Positive emotion|Positive emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin'', ''134''(6), 807–828. https://doi.org/10.1037/a0013341

Campos, B., Shiota, M. N., Keltner, D., Gonzaga, G. C., &amp; Goetz, J. L. (2013). What is shared, what is different? Core relational themes and expressive displays of eight positive emotions. ''Cognition and Emotion'', ''27''(1), 37–52. https://doi.org/10.1080/02699931.2012.683852

Carl, J. R., Soskin, D. P., Kerns, C., &amp; Barlow, D. H. (2013). Positive emotion regulation in emotional disorders: A theoretical review. ''Clinical Psychology Review'', ''33''(3), 343–360. https://doi.org/10.1016/j.cpr.2013.01.003

Fredrickson B. L. (2001). The role of positive emotions in positive psychology. The broaden-and-build theory of positive emotions. ''The American psychologist'', ''56''(3), 218–226. https://doi.org/10.1037//0003-066x.56.3.218

Fredrickson, B. L., &amp; Branigan, C. (2005). Positive emotions broaden the scope of attention and thought‐action repertoires. ''Cognition &amp; Emotion'', ''19''(3), 313–332. https://doi.org/10.1080/02699930441000238

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271

Gruber, J., Mauss, I. B., &amp; Tamir, M. (2011). A Dark Side of Happiness? How, When, and Why Happiness Is Not Always Good. ''Perspectives on Psychological Science'', ''6''(3), 222–233. https://doi.org/10.1177/1745691611406927

Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry'', ''26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Gruber, J., Villanueva, C., Burr, E., Purcell, J. R., &amp; Karoly, H. (2020). Understanding and Taking Stock of Positive Emotion Disturbance. ''Social and personality psychology compass'', ''14''(1), e12515. https://doi.org/10.1111/spc3.12515

Vogel, A. C., Brotman, M. A., Roy, A. K., &amp; Perlman, S. B. (2023). Review: Defining positive emotion dysregulation: Integrating temperamental and clinical perspectives. ''Journal of the American Academy of Child &amp; Adolescent Psychiatry'', ''62''(3), 297–305. https://doi.org/10.1016/j.jaac.2022.06.019

}}

==External links==

* [https://www.psychologytoday.com/au/blog/everyday-resilience/202404/emotional-well-being-5-healthy-practices-for-regulation Emotional Wellbeing] (Psychology Today)
* [https://www.youtube.com/watch?v=MyfzIQH6YKI Positive Emotions with Barbara Fredrickson] (Youtube.com)
* [https://sk.sagepub.com/ency/edvol/the-sage-encyclopedia-of-lifespan-human-development/chpt/reward-sensitivity Reward Sensitivity] (sk.sagepub.com)

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      <text bytes="26023" sha1="qwn23egvqgroxmspj0ff6sb80onuyjk" xml:space="preserve">{{title|Positive emotion dysregulation:&lt;br&gt;What is positive emotion dysregulation and how does it affect psychological functioning?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=8}}
[[File:Woman in red sweater with hand in air.jpg|right|thumb|250px|'''Figure 1.''' Grace experiencing happiness and excitement following several positive life events.]]

'''Scenario'''

Grace is a university student who feels that she has just won the lottery of her life. First, she receives a PhD offer from her dream university. Shortly afterwards, she learns that she has been awarded a scholarship to support her studies. 

For next several days, Grace feels unusually happy and energised (see Figure 1). She sleeps very little because she believes there is too much to achieve. To celebrate her success, she purchases an expensive laptop, makes an impulsive travel booking and commits to several university projects despite having a full schedule. 

People close to Grace begin to wonder if something is not right. Her close friends tried to slow her down, but she ignored them. From Grace's perspective, she is just happy, motivated and confident as pieces of her life are finally coming together in a positive way. 

💡 What is happening to Grace?

💡 Why are her positive emotions become so intense?

{{RoundBoxBottom}}

[[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] are a normal and important part of human life. It includes a variety of feelings such as joy, interest, love, contentment, gratitude, awe, and amusement (Campos et al., 2013). These emotions play an important role in expanding people's thoughts and possible actions to help them build lasting psychological and social resources (Fredickson, 2001). However, positive emotions might not always be beneficial if they are inadequately regulated in terms of magnitude, context, and duration. A review by Gruber et al. (2020) on positive emotion disturbance suggests that extremely strong or prolonged positive emotional states can sometimes interfere with adaptive functioning. Positive emotion dysregulation helps to explain how positive emotional states can feel initially feel good but can sometimes contribute to harmful thoughts and behaviour. This chapter will explore how positive emotions can shift from being adaptive to maladaptive, and how it can affect psychological functioning.  

{{RoundBoxTop|theme=2}}
🔎 '''Focus questions'''

* What is positive emotion dysregulation, and how it can be distinguished from healthy positive emotions?
* What psychological and biological factors contribute to positive emotion dysregulation?
* How does positive emotion dysregulation manifest in thoughts, motivation, and behaviour?
* How can positive emotion dysregulation affect psychological functioning and wellbeing? 

{{RoundBoxBottom}}

==Positive emotion: From benefit to dysregulation==
Understanding positive emotion dysregulation requires more than just identifying whether an emotional experience is pleasant or intense. Positive emotions are usually considered pleasant in nature. But impact of positive emotions will be always relative to how it is integrated into cognitive, behavioural, goal-directed and other factors. This section will consider why positive emotions are important, how they are normally regulated and how difficulties in regulating them can lead to positive emotion dysregulation.  

=== Positive emotions and its adaptive functions ===

* Positive emotions play an important role in psychological functioning. They include numerous experiences such as joy, interest, love, contentment, gratitude, awe, and amusement which can influence cognition, behaviour, and social interaction (Campos et al., 2013). Instead of simply producing pleasant effects, positive emotions can encourage individuals to explore their environment, learn from experiences, engage with others and pursue meaningful goals.
* Positive emotions can broaden people's thoughts, cognition and behaviour. [[wikipedia:Broaden-and-build|Broaden-and-Build Theory]] by Fredrickson (2001) proposes that positive emotions broaden momentary thought-action repertoires that encourages exploration, flexibility and engagement.
* Positive emotions can help to build psychological and social resources, including relationships and coping resources (Fredrickson, 2001).
* Positive emotions are not always beneficial in every circumstances. Effects of positive emotions may depend on intensity, timing, and context. This provides a foundation for understanding when positive emotion may become less adaptive (Gruber et al., 2011).

=== Understanding emotion regulation ===

* Emotional regulation is one’s ability to seek control over own emotional state (Gross, 2015). The process model of emotion regulation by Gross (1998) provides a framework for understanding how individual can modify emotional experiences.
* Positive emotions also require regulation. According to Carl et al., (2013), individuals may need to regulate their positive emotions based on their goals and circumstances. 
* Adaptive emotion regulation is more flexible in nature compared to emotion reduction. Being able to regulate one’s positive emotions can be helpful in ensuring effective functioning.

=== From positive emotions to dysregulation ===

* An intense positive emotion does not necessarily means dysregulated. Feelings such as happiness, excitement, or enthusiasm may be well-placed responses to certain life events.
* Context, intensity, persistence and implications are crucial. Positive emotions can be considered maladaptive if are poorly fitted to the situation or if they start to interfere with functioning (Gruber et al., 2011). 
* The topic of positive emotion dysregulation is a relatively new area of interest. A review by Vogue et al. (2023), suggests that there is a need to examine the challenges of regulating positive emotions rather than focusing exclusively on negative emotions. 

== Mechanisms underlying positive emotion dysregulation ==
 
Positive emotion dysregulation is unlikely result from a single factor. Instead, it is a result of psychological, physiological, biological, individual, and environment factors. All these factors may interact to influence how strongly positive emotions are experienced and how they are regulated.  

{{ic|An alternative approach here could be to focus on the main theory or theories used to understand PED. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt;}}

(Not sure if section is needed to answer the topic question). Might keep it or remove it for book chapter for more clarity. 

Consfused between these two format: 

Format 1: (Focus question 1: What is it?                                               

Focus question 2: How does it manifest?                              

Focus question 3: How does it affect functioning?)                                

Format 2: (Focus question 1: What is it? 

Focus question 2: Why it is happening? 

Focus question 3: How does it manifest? 

Focus question 4: How does it affect functioning?) 

=== Psychological factors ===

* Emotion-regulation processes may influence how positive emotional states are managed. 
* Reward sensitivity may influence responses to positive experiences.
* Motivational and cognitive processes may shape the results of positive emotion.

=== Physiological and biological factors ===

* Positive emotions include measurable physiological and neural responses.
* Biological reward system may contribute to positive emotional responding.
* Little research available in the field of positive emotion dysregulation.

=== Individual and environmental factors ===

* Temperament may influence positive emotional reactivity (Vogel et al., 2023).
* Individual and contextual factors influences emotion regulation. 
* Situation can determine whether positive emotional activation is adaptive. 

== Consequences of positive emotion dysregulation ==

The regulation of positive emotions may manifest itself through altered thinking processes, decision-making, goals pursued and behaviour of an individual. While positive emotions helps to expand attention and motivate individuals to explore more, strong and poorly regulated emotional activation may influence judgement. This section will discuss the way in which positive experiences may impact cognition, motivation and decision-making processes.

=== Changes in cognition and decision-making ===

* Positive emotions can broaden attention and action thought repertoires. Fredrickson and Branigan (2005) found that positive affective states broadened both attention and the range of thought-action responses that the participants gave. It provided empirical support for broaden-and-build theory. 
* Positive emotion dysregulation can influence decision-making.
* The effects of cognitive broadening are context-dependent 

=== Goal pursuit and behavioural activation ===

* The positive emotions would enhance approach motivation and behavioural activation. 
* Excessive activation might help achieve goals but could be exaggerated. In case of exaggeration, the activation might make people over-commit.
* Grace’s behaviour explains this as initially she was motivated to participate in university opportunities. But repeatedly taking additional projects despite limited time shows that goal directed activation is becoming difficult to regulate. 

=== Risk-taking and Impulsivity ===

* Presence of positive emotional activation may lead to increased level of impulsivity and risky behaviour. 
* Positive urgency provides a precise reason for rash behaviour during high levels of positive emotions. Cyders and Smith (2008) define positive urgency as the tendency of behaving rashly during high levels of positive affect. This behavioural trend can be risky as well. 
* Impulsivity and risk-taking should not be equated with positive emotion dysregulation  

== Positive emotion dysregulation and psychological functioning ==
{{expand}}

=== Positive emotions as good psychological resources ===

* Positive emotions can support engagement and goal-directed functioning.
* Valuable for academic achievement.
* Encourage exploration and persistence.

=== When positive emotional experiences becomes difficult to regulate ===

* Poorly regulated positive emotions can create interpersonal difficulties.
* Positive emotion dysregulation may interfere with everyday responsibilities.
* It can affect sleep and concentration 

;Quiz
Choose your answer and click &quot;Submit&quot;
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement explains why positive emotion dysregulation can affect psychological functioning?
|type=&quot;()&quot;}
+ Poorly regulated positive emotion may interfere with wellbeing, relationships and responsibilities
- Positive emotions are always harmful
- Positive emotions have no effect on behaviour
- Intense positive emotions always indicate dysregulation

&lt;/quiz&gt;

==Conclusion==
*Positive emotion dysregulation is not simply experiencing intense happiness, but difficulty in regulating positive emotions.
*Dysregulation can influence thoughts, decision-making and behaviour.
*Difficulty in regulating positive emotions may affect everyday responsibilities and functioning. 

==See also ==
*[[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory of positive emotions]] (Book chapter, 2016)
*[[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion Dysregulation]] (Book chapter, 2026)
*[[wikipedia:Emotional_dysregulation|Emotional Dysregulation]] (Wikipedia)
*[[Motivation and emotion/Book/2020/Positive emotion|Positive emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin'', ''134''(6), 807–828. https://doi.org/10.1037/a0013341

Campos, B., Shiota, M. N., Keltner, D., Gonzaga, G. C., &amp; Goetz, J. L. (2013). What is shared, what is different? Core relational themes and expressive displays of eight positive emotions. ''Cognition and Emotion'', ''27''(1), 37–52. https://doi.org/10.1080/02699931.2012.683852

Carl, J. R., Soskin, D. P., Kerns, C., &amp; Barlow, D. H. (2013). Positive emotion regulation in emotional disorders: A theoretical review. ''Clinical Psychology Review'', ''33''(3), 343–360. https://doi.org/10.1016/j.cpr.2013.01.003

Fredrickson B. L. (2001). The role of positive emotions in positive psychology. The broaden-and-build theory of positive emotions. ''The American psychologist'', ''56''(3), 218–226. https://doi.org/10.1037//0003-066x.56.3.218

Fredrickson, B. L., &amp; Branigan, C. (2005). Positive emotions broaden the scope of attention and thought‐action repertoires. ''Cognition &amp; Emotion'', ''19''(3), 313–332. https://doi.org/10.1080/02699930441000238

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271

Gruber, J., Mauss, I. B., &amp; Tamir, M. (2011). A Dark Side of Happiness? How, When, and Why Happiness Is Not Always Good. ''Perspectives on Psychological Science'', ''6''(3), 222–233. https://doi.org/10.1177/1745691611406927

Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry'', ''26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Gruber, J., Villanueva, C., Burr, E., Purcell, J. R., &amp; Karoly, H. (2020). Understanding and Taking Stock of Positive Emotion Disturbance. ''Social and personality psychology compass'', ''14''(1), e12515. https://doi.org/10.1111/spc3.12515

Vogel, A. C., Brotman, M. A., Roy, A. K., &amp; Perlman, S. B. (2023). Review: Defining positive emotion dysregulation: Integrating temperamental and clinical perspectives. ''Journal of the American Academy of Child &amp; Adolescent Psychiatry'', ''62''(3), 297–305. https://doi.org/10.1016/j.jaac.2022.06.019

}}

==External links==

* [https://www.psychologytoday.com/au/blog/everyday-resilience/202404/emotional-well-being-5-healthy-practices-for-regulation Emotional Wellbeing] (Psychology Today)
* [https://www.youtube.com/watch?v=MyfzIQH6YKI Positive Emotions with Barbara Fredrickson] (Youtube.com)
* [https://sk.sagepub.com/ency/edvol/the-sage-encyclopedia-of-lifespan-human-development/chpt/reward-sensitivity Reward Sensitivity] (sk.sagepub.com)
{{title|Title goes here:&lt;br&gt;Subtitle goes here?}}
&lt;div align=center&gt;Edit the wording (and [[w:Stylistic or specialised usage|casing]]) above so that it matches the [[Motivation and emotion/Book/Current|topic list]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not add your name; authorship is shown in the [[Special:History/{{PAGENAME}}|page history]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|200px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Introduce the topic with a scenario

Begin with an engaging scenario, example, or case study that illustrates the topic and gives readers a reason to care about it.

The scenario should:
* Start with a lead in bold, such as '''Scenario, Case study, Imagine this ...''', or another another phrase that suits the scenario.
* Describe a '''realistic problem, situation, or question''' related to the topic.
* Be engaging and accessible to a reader who is new to the topic.
* Provide a context that can be revisited when explaining the psychological concepts and research later in the chapter.
* Avoid explaining theory or research in detail— the purpose is to illustrate the problem, not solve it.
* Be presented in a [[#Feature box|feature box]].
* Include a relevant image, with a figure caption, to help illustrate the scenario. Cite the figure (e.g., see Figure 1) within the scenario.

For the [[Motivation and emotion/Assessment/Topic|topic development]], the scenario can be planned using bullet-points.

;Feature box colour
To change the feature-box colour:
# Select Edit source
# Find theme=3
# Change 3 to another theme number

{{RoundBoxBottom}}

The Overview section should consist of three parts:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Brief explanation of the problem, why it is important, and an outline of how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.

This template provides key headings, examples, and tips for each section. Gradually remove this generic information as the chapter develops. It is OK to retain some template material for the topic development, but it should all be removed from the book chapter.

The topic development submission should communicate your current thinking and plans for the project. It is not expected to be a fully developed or final product.

Key resources:
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] explains how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] questions. For example:

{{cross}} Is there a relationship between weather and criminal behaviour? (closed-ended)&lt;br&gt;
{{tick}} What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Headings==
Each chapter should use this standard heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Emotion regulation]]
[[Category:Motivation and emotion/Book/Positive emotion]]</text>
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        <username>P U3270518</username>
        <id>3106535</id>
      </contributor>
      <comment>Focus questions. Copilot was used to make focus question more precise and grammatically correct. Link to the chat: https://copilot.cloud.microsoft/chat/conversation/5062d1b6-42b9-4233-863e-27e249cab315?fromcode=cmmqbg1y3ah&amp;es=UnauthClick&amp;login_hint=u3270518%40uni.canberra.edu.au</comment>
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      <text bytes="26031" sha1="nbd9vqegw8pcttrc9pfbzr5kpwp7g1k" xml:space="preserve">{{title|Positive emotion dysregulation:&lt;br&gt;What is positive emotion dysregulation and how does it affect psychological functioning?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=8}}
[[File:Woman in red sweater with hand in air.jpg|right|thumb|250px|'''Figure 1.''' Grace experiencing happiness and excitement following several positive life events.]]

'''Scenario'''

Grace is a university student who feels that she has just won the lottery of her life. First, she receives a PhD offer from her dream university. Shortly afterwards, she learns that she has been awarded a scholarship to support her studies. 

For next several days, Grace feels unusually happy and energised (see Figure 1). She sleeps very little because she believes there is too much to achieve. To celebrate her success, she purchases an expensive laptop, makes an impulsive travel booking and commits to several university projects despite having a full schedule. 

People close to Grace begin to wonder if something is not right. Her close friends tried to slow her down, but she ignored them. From Grace's perspective, she is just happy, motivated and confident as pieces of her life are finally coming together in a positive way. 

💡 What is happening to Grace?

💡 Why are her positive emotions become so intense?

{{RoundBoxBottom}}

[[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] are a normal and important part of human life. It includes a variety of feelings such as joy, interest, love, contentment, gratitude, awe, and amusement (Campos et al., 2013). These emotions play an important role in expanding people's thoughts and possible actions to help them build lasting psychological and social resources (Fredickson, 2001). However, positive emotions might not always be beneficial if they are inadequately regulated in terms of magnitude, context, and duration. A review by Gruber et al. (2020) on positive emotion disturbance suggests that extremely strong or prolonged positive emotional states can sometimes interfere with adaptive functioning. Positive emotion dysregulation helps to explain how positive emotional states can feel initially feel good but can sometimes contribute to harmful thoughts and behaviour. This chapter will explore how positive emotions can shift from being adaptive to maladaptive, and how it can affect psychological functioning.  

{{RoundBoxTop|theme=2}}
🔎 '''Focus questions'''

1️⃣ What is positive emotion dysregulation, and how it can be distinguished from healthy positive emotions?

2️⃣ What factors contribute to difficulties in regulating positive emotions? 

3️⃣ How are positive emotions regulated, and when can regulation become maladaptive? 

4️⃣  How does positive emotion dysregulation affect psychological functioning and wellbeing? 

{{RoundBoxBottom}}

==Positive emotion: From benefit to dysregulation==
Understanding positive emotion dysregulation requires more than just identifying whether an emotional experience is pleasant or intense. Positive emotions are usually considered pleasant in nature. But impact of positive emotions will be always relative to how it is integrated into cognitive, behavioural, goal-directed and other factors. This section will consider why positive emotions are important, how they are normally regulated and how difficulties in regulating them can lead to positive emotion dysregulation.  

=== Positive emotions and its adaptive functions ===

* Positive emotions play an important role in psychological functioning. They include numerous experiences such as joy, interest, love, contentment, gratitude, awe, and amusement which can influence cognition, behaviour, and social interaction (Campos et al., 2013). Instead of simply producing pleasant effects, positive emotions can encourage individuals to explore their environment, learn from experiences, engage with others and pursue meaningful goals.
* Positive emotions can broaden people's thoughts, cognition and behaviour. [[wikipedia:Broaden-and-build|Broaden-and-Build Theory]] by Fredrickson (2001) proposes that positive emotions broaden momentary thought-action repertoires that encourages exploration, flexibility and engagement.
* Positive emotions can help to build psychological and social resources, including relationships and coping resources (Fredrickson, 2001).
* Positive emotions are not always beneficial in every circumstances. Effects of positive emotions may depend on intensity, timing, and context. This provides a foundation for understanding when positive emotion may become less adaptive (Gruber et al., 2011).

=== Understanding emotion regulation ===

* Emotional regulation is one’s ability to seek control over own emotional state (Gross, 2015). The process model of emotion regulation by Gross (1998) provides a framework for understanding how individual can modify emotional experiences.
* Positive emotions also require regulation. According to Carl et al., (2013), individuals may need to regulate their positive emotions based on their goals and circumstances. 
* Adaptive emotion regulation is more flexible in nature compared to emotion reduction. Being able to regulate one’s positive emotions can be helpful in ensuring effective functioning.

=== From positive emotions to dysregulation ===

* An intense positive emotion does not necessarily means dysregulated. Feelings such as happiness, excitement, or enthusiasm may be well-placed responses to certain life events.
* Context, intensity, persistence and implications are crucial. Positive emotions can be considered maladaptive if are poorly fitted to the situation or if they start to interfere with functioning (Gruber et al., 2011). 
* The topic of positive emotion dysregulation is a relatively new area of interest. A review by Vogue et al. (2023), suggests that there is a need to examine the challenges of regulating positive emotions rather than focusing exclusively on negative emotions. 

== Mechanisms underlying positive emotion dysregulation ==
 
Positive emotion dysregulation is unlikely result from a single factor. Instead, it is a result of psychological, physiological, biological, individual, and environment factors. All these factors may interact to influence how strongly positive emotions are experienced and how they are regulated.  

{{ic|An alternative approach here could be to focus on the main theory or theories used to understand PED. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt;}}

(Not sure if section is needed to answer the topic question). Might keep it or remove it for book chapter for more clarity. 

Consfused between these two format: 

Format 1: (Focus question 1: What is it?                                               

Focus question 2: How does it manifest?                              

Focus question 3: How does it affect functioning?)                                

Format 2: (Focus question 1: What is it? 

Focus question 2: Why it is happening? 

Focus question 3: How does it manifest? 

Focus question 4: How does it affect functioning?) 

=== Psychological factors ===

* Emotion-regulation processes may influence how positive emotional states are managed. 
* Reward sensitivity may influence responses to positive experiences.
* Motivational and cognitive processes may shape the results of positive emotion.

=== Physiological and biological factors ===

* Positive emotions include measurable physiological and neural responses.
* Biological reward system may contribute to positive emotional responding.
* Little research available in the field of positive emotion dysregulation.

=== Individual and environmental factors ===

* Temperament may influence positive emotional reactivity (Vogel et al., 2023).
* Individual and contextual factors influences emotion regulation. 
* Situation can determine whether positive emotional activation is adaptive. 

== Consequences of positive emotion dysregulation ==

The regulation of positive emotions may manifest itself through altered thinking processes, decision-making, goals pursued and behaviour of an individual. While positive emotions helps to expand attention and motivate individuals to explore more, strong and poorly regulated emotional activation may influence judgement. This section will discuss the way in which positive experiences may impact cognition, motivation and decision-making processes.

=== Changes in cognition and decision-making ===

* Positive emotions can broaden attention and action thought repertoires. Fredrickson and Branigan (2005) found that positive affective states broadened both attention and the range of thought-action responses that the participants gave. It provided empirical support for broaden-and-build theory. 
* Positive emotion dysregulation can influence decision-making.
* The effects of cognitive broadening are context-dependent 

=== Goal pursuit and behavioural activation ===

* The positive emotions would enhance approach motivation and behavioural activation. 
* Excessive activation might help achieve goals but could be exaggerated. In case of exaggeration, the activation might make people over-commit.
* Grace’s behaviour explains this as initially she was motivated to participate in university opportunities. But repeatedly taking additional projects despite limited time shows that goal directed activation is becoming difficult to regulate. 

=== Risk-taking and Impulsivity ===

* Presence of positive emotional activation may lead to increased level of impulsivity and risky behaviour. 
* Positive urgency provides a precise reason for rash behaviour during high levels of positive emotions. Cyders and Smith (2008) define positive urgency as the tendency of behaving rashly during high levels of positive affect. This behavioural trend can be risky as well. 
* Impulsivity and risk-taking should not be equated with positive emotion dysregulation  

== Positive emotion dysregulation and psychological functioning ==
{{expand}}

=== Positive emotions as good psychological resources ===

* Positive emotions can support engagement and goal-directed functioning.
* Valuable for academic achievement.
* Encourage exploration and persistence.

=== When positive emotional experiences becomes difficult to regulate ===

* Poorly regulated positive emotions can create interpersonal difficulties.
* Positive emotion dysregulation may interfere with everyday responsibilities.
* It can affect sleep and concentration 

;Quiz
Choose your answer and click &quot;Submit&quot;
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement explains why positive emotion dysregulation can affect psychological functioning?
|type=&quot;()&quot;}
+ Poorly regulated positive emotion may interfere with wellbeing, relationships and responsibilities
- Positive emotions are always harmful
- Positive emotions have no effect on behaviour
- Intense positive emotions always indicate dysregulation

&lt;/quiz&gt;

==Conclusion==
*Positive emotion dysregulation is not simply experiencing intense happiness, but difficulty in regulating positive emotions.
*Dysregulation can influence thoughts, decision-making and behaviour.
*Difficulty in regulating positive emotions may affect everyday responsibilities and functioning. 

==See also ==
*[[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory of positive emotions]] (Book chapter, 2016)
*[[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion Dysregulation]] (Book chapter, 2026)
*[[wikipedia:Emotional_dysregulation|Emotional Dysregulation]] (Wikipedia)
*[[Motivation and emotion/Book/2020/Positive emotion|Positive emotion]] (Book chapter, 2020)

==References==
{{Hanging indent|1=

Cyders, M. A., &amp; Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin'', ''134''(6), 807–828. https://doi.org/10.1037/a0013341

Campos, B., Shiota, M. N., Keltner, D., Gonzaga, G. C., &amp; Goetz, J. L. (2013). What is shared, what is different? Core relational themes and expressive displays of eight positive emotions. ''Cognition and Emotion'', ''27''(1), 37–52. https://doi.org/10.1080/02699931.2012.683852

Carl, J. R., Soskin, D. P., Kerns, C., &amp; Barlow, D. H. (2013). Positive emotion regulation in emotional disorders: A theoretical review. ''Clinical Psychology Review'', ''33''(3), 343–360. https://doi.org/10.1016/j.cpr.2013.01.003

Fredrickson B. L. (2001). The role of positive emotions in positive psychology. The broaden-and-build theory of positive emotions. ''The American psychologist'', ''56''(3), 218–226. https://doi.org/10.1037//0003-066x.56.3.218

Fredrickson, B. L., &amp; Branigan, C. (2005). Positive emotions broaden the scope of attention and thought‐action repertoires. ''Cognition &amp; Emotion'', ''19''(3), 313–332. https://doi.org/10.1080/02699930441000238

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271

Gruber, J., Mauss, I. B., &amp; Tamir, M. (2011). A Dark Side of Happiness? How, When, and Why Happiness Is Not Always Good. ''Perspectives on Psychological Science'', ''6''(3), 222–233. https://doi.org/10.1177/1745691611406927

Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry'', ''26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781

Gruber, J., Villanueva, C., Burr, E., Purcell, J. R., &amp; Karoly, H. (2020). Understanding and Taking Stock of Positive Emotion Disturbance. ''Social and personality psychology compass'', ''14''(1), e12515. https://doi.org/10.1111/spc3.12515

Vogel, A. C., Brotman, M. A., Roy, A. K., &amp; Perlman, S. B. (2023). Review: Defining positive emotion dysregulation: Integrating temperamental and clinical perspectives. ''Journal of the American Academy of Child &amp; Adolescent Psychiatry'', ''62''(3), 297–305. https://doi.org/10.1016/j.jaac.2022.06.019

}}

==External links==

* [https://www.psychologytoday.com/au/blog/everyday-resilience/202404/emotional-well-being-5-healthy-practices-for-regulation Emotional Wellbeing] (Psychology Today)
* [https://www.youtube.com/watch?v=MyfzIQH6YKI Positive Emotions with Barbara Fredrickson] (Youtube.com)
* [https://sk.sagepub.com/ency/edvol/the-sage-encyclopedia-of-lifespan-human-development/chpt/reward-sensitivity Reward Sensitivity] (sk.sagepub.com)
{{title|Title goes here:&lt;br&gt;Subtitle goes here?}}
&lt;div align=center&gt;Edit the wording (and [[w:Stylistic or specialised usage|casing]]) above so that it matches the [[Motivation and emotion/Book/Current|topic list]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not add your name; authorship is shown in the [[Special:History/{{PAGENAME}}|page history]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|200px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Introduce the topic with a scenario

Begin with an engaging scenario, example, or case study that illustrates the topic and gives readers a reason to care about it.

The scenario should:
* Start with a lead in bold, such as '''Scenario, Case study, Imagine this ...''', or another another phrase that suits the scenario.
* Describe a '''realistic problem, situation, or question''' related to the topic.
* Be engaging and accessible to a reader who is new to the topic.
* Provide a context that can be revisited when explaining the psychological concepts and research later in the chapter.
* Avoid explaining theory or research in detail— the purpose is to illustrate the problem, not solve it.
* Be presented in a [[#Feature box|feature box]].
* Include a relevant image, with a figure caption, to help illustrate the scenario. Cite the figure (e.g., see Figure 1) within the scenario.

For the [[Motivation and emotion/Assessment/Topic|topic development]], the scenario can be planned using bullet-points.

;Feature box colour
To change the feature-box colour:
# Select Edit source
# Find theme=3
# Change 3 to another theme number

{{RoundBoxBottom}}

The Overview section should consist of three parts:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Brief explanation of the problem, why it is important, and an outline of how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.

This template provides key headings, examples, and tips for each section. Gradually remove this generic information as the chapter develops. It is OK to retain some template material for the topic development, but it should all be removed from the book chapter.

The topic development submission should communicate your current thinking and plans for the project. It is not expected to be a fully developed or final product.

Key resources:
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] explains how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] questions. For example:

{{cross}} Is there a relationship between weather and criminal behaviour? (closed-ended)&lt;br&gt;
{{tick}} What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Headings==
Each chapter should use this standard heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Emotion regulation]]
[[Category:Motivation and emotion/Book/Positive emotion]]</text>
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== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]]

;Scenario

Ash is a swimmer training for a national competition. Being a swimmer is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00am every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle?
{{RoundBoxBottom}}

Your self-concept is your understanding of who you are and how you feel about yourself. This includes a complex system of the attitudes, beliefs and judgements you hold about yourself. Self-concept guides you in answering the question “who am I?” (Wehrle &amp; Fasbender, 2018). But its importance extends beyond answering that question. Self-concept influences how you think, feel and behave. Consider self-concept an internal compass which motivates and guides your choices, relationships and understanding of the world.  

For Ash in the case study, being a swimmer and succeeding in the upcoming swimming competition are core to her self-concept. This guides and motivates her behaviour of swimming early every morning.  

This chapter explores how our understanding of ourselves influences what we’re motivated to do and examines how what we repeatedly do subsequently shapes our self-concept. By understanding how and why we perceive ourselves the way we do, and the impacts of our self-concepts, we can better understand our behaviours, foster positive self-concept and embrace the motivational properties of self-concept.{{RoundBoxTop|theme=3}}
'''Focus questions'''

*What is self-concept? 
*What are the key theories of self-concept? 
*How does self-concept influence motivation? 
*How does motivation influence self-concept?
*How can our understanding of self-concept be used to improve motivation and wellbeing?{{RoundBoxBottom}}

== Understanding self-concept ==
The self is central to humanhood and the scientific understanding of human behaviour. Thus, the question of “what makes me, me?” has been theorised about throughout history and is important to understand. This section details the core theories and properties of self-concept which will be referred to throughout the chapter.  

Self-concept is a cognitive framework which is unique to, but encompasses, many of the popular “self-” constructs – [[self-esteem]] (perceived value of yourself), [[w:Self-efficacy|self-efficacy]] (perceived task capability) and [[w:Self-image|self-image]] (description of yourself). See the linked pages for details on each of these separate constructs. 

==== William James ====
James (1892) proposed the self consists of two parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. This philosophical framework laid the basis for the development of empirical theories of self-concept and catapulted psychological identity research.  

==== The looking-glass self ====
Cooley (1902) framed the individual as a social being, theorising that self-concept is a reflection of the responses and evaluations of individuals within one’s environment. Self-concept is developed through the imagined judgement of others, and individuals become the people others say they are (Siljanovska &amp; Stojcevska, 2018). 

==== Rogers’ self theory ====
Rogers (1959), a humanist psychologist, proposed self-concept is a dynamic, organised system which consists of three parts: the real self (how you see yourself), ideal self (who you would like to be), and self-worth (how much you value yourself). He proposed that individuals behave in accordance with their self-concept, a stepping stone toward the contemporary understanding of how self-beliefs influence motivation (Ismail &amp; Tekke, 2015; Yousefi &amp; Kiani, 2014).  

==== The multidimensional self-concept model ====
Marsh et al. (1992) were critical of the lack of theoretical and methodological rigor supporting studies of self-concept. They built upon previous theories, suggesting self-concept is the perception of oneself shaped by experience and evaluations, but proposed self-concept is organised, multifaceted and hierarchical (Marsh et al., 2019). An individual’s self-concept is comprised of academic, social, emotional and physical dimensions which each divide into further subsets of increasing specificity and are closer related to actual behaviour (see Figure 2). This original model was later refined by developing and analysing the Self Description Questionnaire and produced a model which captures the correlations between each aspect of self-concept (see Figure 2; Marsh &amp; Shavelson, 1985). This hierarchical model predicts behaviour and feelings more accurately than previous general theories (Crocker et al., 2006; Marsh et al., 2019).  [[File:Multidimensional model of self-concept diagram.png|thumb|600x600px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 1992).|center]]
==== Properties of self-concept   ====
Although these theories each approach self-concept differently, several common properties emerge across them. Self-concept appears to be a multifaceted, hierarchical, socially developed, and dynamic system of self-knowledge (Table 1). These properties provide the foundation for understanding the motivational effects of self-concept. 

'''Table 1'''

Summary Table of Self-Concept Properties and Their Significance for Motivation.
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
!Property
!Theory
!Significance for motivation
|-
|'''Multifaceted'''
|James, Rogers, Multidimensional model
|Different aspects of the self motivate different behaviours
|-
|'''Hierarchical  '''
|Multidimensional model
|General and specific domains influence each other and different behaviours
|-
|'''Socially developed'''
|The looking glass self
|Social feedback influences which behaviours are pursued  
|-
|'''Dynamic'''
|Rogers, Multidimensional model
|Self-concept changes with experience
|-
|'''Evaluative'''
|Rogers
|Perceived self-worth affects responses to success/failure
|-
|'''Descriptive  '''
|Rogers, Multidimensional model
|Identity beliefs influence goals and expectations
|}

== The influence of self-concept on motivation ==
'''''[Author's note: I have begun drafting out the chapter in full-sentence form. I am keeping my planned key points for each''''' '''''of the following sections in italics.]''''' 

* ''A growing body of research suggests self-concept has important motivational properties.'' 

* ''Motivation is an internal process which initiates, directs and sustains behaviour (Lens &amp; Vansteenkiste, 2020).''

There is a growing body of research suggesting self-concept has important motivational properties. Motivation is an internal process which initiates, directs and sustains behaviour (Lens &amp; Vansteenkiste, 2020). Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation are both central guiding forces of behaviour and navigating the environment. This section explores how self-concept influences motivation.  

=== How does self-concept direct goals? ===

* ''Rogers’ (1959) congruence theory explains people are driven to behave in ways to conform to their real and ideal selves.''

* ''Empirical testing of congruence produces mixed results (Boldero &amp; Francis, 2002; Howard &amp; Dhillon, 2022; Phillips et al., 1965).'' 

* ''Oyserman’s (2024) identity-based motivation theory provides a more flexible account of how self-concept influences behaviour in everyday situations, suggesting identities are contextually salient.  ''

* ''When behaviour is congruent with one’s identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). Incongruent actions that have the same challenges are interpreted as pointless. Identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024).'' 

* ''Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, increased when children were guided to connect with their future selves.'' 

* ''Self-concept directs behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted.'' 

Individuals pursue and evaluate goals in ways that align with their self-concept. This is driven by the human tendency to strive for alignment between self-concept and behaviour.  

'''Rogers’ (1959) theory''' suggests the self is comprised of the real self and the ideal self. The alignment of these is known as congruence, a state where one has the capacity to pursue self-improving goals. When the ideal and real self do not align (incongruence), an individual may experience poorer mental health and act defensively. This incongruence generates a psychological tension which acts as a motivating force, pushing the individual to either modify their behaviour to align with their self-concept, or modify their self-concept (Stephen, 2023).  

Supporting this concept, later research has similarly suggested that ideal standards function as reference values individuals use to evaluate their current behaviour and direct goal pursuit (Boldero &amp; Francis, 2002). Howard and Dhillon (2022) also found using Rogers’ Q-sort, a measure of self-concept and congruency, can be an effective method of reflection to help individuals consciously choose goals that help achieve a congruent, ideal self.  

However, direct support for the foundational concept of congruence is sparse (Phillips et al., 1965). Rogers’ Q-sort was used to empirically test the concept of congruence. Phillips et al. (1965) found that incongruent participants, as indicated by Q-sort results, did not become more congruent over a 6-month measurement period, thus indicating a lack of movement motivated by incongruency. One explanation is that Rogers conceptualised self-concept as relatively stable. But this framework has not been fully discounted. In response to the poor empirical support for congruence, contemporary research tends to adopt a dynamic and context dependent approach to self-concept. Oyserman’s (2024) identity-based motivation theory, for example considers self-concept as continually constructed dynamically and contextually. This approach may therefore provide a more flexible and realistic account of how self-concept influences behaviour in everyday situations. 

'''[[wikipedia:Identity_based_motivation|Identity-based motivation theory]]''' suggests situations influence which aspects of our multidimensional self-concept are active – the most salient identity on one’s mind - and thus influence behaviour. Like Rogers’ theory, it suggests individuals behave and interpret situations in ways that align with their identity. However, Oyserman (2015) expands Roger’s simplistic differentiation between real and ideal selves. The theory suggests a temporal flexibility of self-concept. A future self can be an active identity and thus individuals may act congruently to future selves. This may explain why future identities motivate present behaviour. Unlike Rogers' relatively stable model of the ideal self, identity-based motivation theory suggests future identities can become situationally salient and influence immediate behavioural decisions. 

The theory suggests that when behaviour is congruent with one’s identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). Incongruent actions that have the same challenges are interpreted as pointless. Identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). 

But why is importance motivating? Identity-relevant goals are motivating because they provide information about who one is and who one wants to become. This makes difficulty meaningful rather than discouraging (Oyserman, 2024).
{{RoundBoxTop|theme=3}}'''Case study comprehension check:'''
Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}}

Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children and subsequent achievement increased when children were guided to connect with their future selves. In two separate studies, students were primed to think of their adult-self as near or far. In the first study, those in the adult-self is near group performed significantly better on a geometry task than the control and far group. The second study used a longer-term measure of final grades. Students in the adult-self is near group again performed significantly better than the far group. This longer-term effect of connecting to one’s future self lasted six months. These findings indicate that identifying with one’s future/ideal self can motivate goal attainment. Self-concept is malleable and moderates goal pursuit based on how one perceives themselves at a given time.  

Together, these theories suggest self-concept influences motivation through two complementary processes. Individuals are motivated to reduce discrepancies between their real and ideal selves. Additionally, individuals are motivated to pursue goals that are congruent with situationally active identities. Thus, self-concept directs behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted.  

=== What is the role of perceived competence in the effect of self-concept on motivation? ===

* ''Evaluative beliefs about competence are part of self-concept. Individuals engage with goals they believe they are more capable of achieving (Marsh et al., 2019).'' 

* ''Seaton et al.'s (2014) study found prior mathematics self-concept was a significant positive predictor of mastery-approach and a positive, but weaker predictor of performance-approach goal orientations.'' 

* ''Goal orientation theory: mastery goal approaches involve being motivated to learn, whereas performance goal approaches are motivated by wanting to demonstrate competence (Vandewalle et al., 2019).'' 

* ''Perceived competence provides a motivational framework for how people engage with goals.'' 

Self-concept incorporates evaluative beliefs about competence which influence how individuals approach goals. Individuals engage with goals they believe they are more capable of achieving (Marsh et al., 2019). For example, Seaton et al.'s (2014) longitudinal study of 2786 Australian students' found prior mathematics self-concept was a significant positive predictor of mastery-approach and a positive, but weaker predictor of performance-approach goal orientations. According to '''[[wikipedia:Goal_orientation|goal orientation theory]]''', a mastery goal approach involves being motivated to learn, whereas a performance goal approach is motivated by wanting to demonstrate competence (Vandewalle et al., 2019). These results, therefore, suggest that domain-specific self-concepts provide a motivational foundation for how students engage with learning tasks. When students perceive themselves as competent, they are more likely to be motivated to learn, rather than merely demonstrate their competence to others.  

Frade and Viega's (2017) study of navy trainees corroborated these results with non-domain specific measures and a Portuguese adult population, indicating results are generalisable beyond Seaton et al.'s (2014) Australian high school participants and the domain of mathematics. However, Seaton et al.'s (2014) study did not incorporate avoidance (goals motivated to prevent failure) orientations of mastery and performance goals in their analyses, limiting the deductive power of these findings and possibly understating the strength of the identified correlation. Nevertheless, results support that perceived self-competence influences goal engagement.  

=== How can self-concept undermine motivation? ===

* ''Self-worth contingencies refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). Holding contingent beliefs about one's value can undermine motivation.'' 

* ''Highly contingent self-worth means failing threatens individuals’ identity.'' 

* ''Lietz et al. (2026) found that high contingent self-worth correlated with participants' negative and severe emotional responses to failure. Participants attributed failure to themselves.''

* ''These finding are consistent with attribution theory (Weiner, 1985): causal thinking processes where people attribute outcomes to stable, uncontrollable factors, results in low motivation. Self-attributions of failure can undermine motivation.'' 

* ''Contingent self-worth is temporary (Fairlamb, 2020; Lawrence &amp; Gonzales, 2022). This can be explained by regulation types detailed in self-determination theory (Ryan &amp; Deci, 2020).'' 

* ''Self-handicapping: creating obstacles to your own goal so that failure cannot be self-attributed (Török et al., 2018).'' 

* ''Rhodewalt and Fairfield (1991) found that participants who scored highly on a measure of self-handicapping tendency and reported intentions to withhold effort in a test subsequently performed worse than others.'' 

* ''Self-concept can act as a motivational vulnerability as well as a motivational resource.''

The theories and research discussed so far optimistically discuss how self-concept motivates one to meet their ideal self. However, there are circumstances under which self-concept undermines motivation. Contingent self-evaluations and self-handicapping help us understand how self-concept can undermine motivation. 

'''[[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]]''' refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). Holding contingent beliefs about one's value can undermine motivation. For individuals who have highly contingent self-worth, achieving goals is closely tied to their self-concept, meaning failure is threatening to one's self-concept. Accordingly, contingent self-worth motivates pursuit of success for domains that are important for one's self-worth. This is consistent with Oyserman and Rogers' theories that individuals tend to set self-consistent goals, and that these goals are highly valued and thus highly motivated. 

Lietz et al. (2026) corroborates that this motivating guide provided by self-concept resulted in participants' negative and severe emotional responses to failure. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept, which resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with '''[[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]]''' (Weiner, 1985), which posits that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). 

Presenting another pitfall of self-concept driven motivation, Fairlamb (2020) suggests contingent self-worth motivation is temporary. This temporary nature is illustrated by Lawrence and Gonzales' (2022) results from 466 university students, wherein self-report measures of motivation and self-worth showed a correlation between academically contingent self-worth and self-worth boosting goals such as studying to graduate, rather than studying for the joy of it. They also showed a correlation between academically contingent self-worth and amotivation, a lack of purpose and drive.  

van der Kaap-Deeder et al. (2016) explains the motivational boost from contingent self-worth is temporary because it is driven by a 'need to do well' motivation (introjected regulation), rather than intrinsic regulation (internal drive). These different motivation types are proposed by '''[[self-determination theory]]''', wherein intrinsic motivation provides a more sustainable motivational force, as the behaviour is powered by its inherent satisfaction (Zhang et al., 2016). Introjected motivation, however, is driven by internal guilt or pressure, resulting in stress and a feeling of a lack of autonomy (Taris et al., 2020). These uncomfortable affective experiences, paired with any perceived risk of failure causes motivation from contingent self-worth to decrease, and maladaptive behaviours emerge (van der Kaap-Deeder et al., 2016). 

One such response to the threat of failure is self-handicapping. '''[[w:Self-handicapping|Self-handicapping]]''' is the act of creating obstacles to one's own goal so that failure may not be self-attributed (Török et al., 2018). Creating a handicap generates an equally plausible and external explanation for one's outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, reflecting the motivating quality of self-concept and indicating self-concept can hinder goal pursuit (Coudevylle et al., 2020). 

{{RoundBoxTop|theme=3}}'''Case study comprehension check:'''
Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence.  {{RoundBoxBottom}}

Schwinger et al. (2022) found a medium negative correlation between general self-esteem and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. Self-esteem mirrors the evaluative property of self-concept; fear of failure reflects concern about the implications of failure for one's self-concept. This highlights self-concept as an antecedent to self-handicapping and related motivational and performance outcomes. In this study, general, rather than domain-specific measures were used. As the multidimensional theory of self-concept suggests, the more specific the self-concept facet, the more closely related to actual behaviour it is. Accordingly, future research should incorporate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation.

Self-handicapping demonstrates that motivation is not always directed towards achieving valued outcomes. Under some circumstances, protecting self-worth becomes more important than succeeding at the task itself. To illustrate the behavioural effects of self-handicapping, Rhodewalt and Fairfield (1991) found that participants who scored highly on a measure of self-handicapping tendency and reported intentions to withhold effort in a test subsequently performed worse than others. This reflects the influence of self-concept on motivation; preserving one’s image undermines motivation to succeed due to the threat of failure and what it means for one’s self concept. Whereas Rogers (1959) and Oyserman (2015) suggest people are motivated to pursue identity-congruent goals, self-handicapping highlights a potential downside of identity-relevant goals. When success becomes central to the self, individuals may become motivated to protect their self-concept rather than maximise performance. 

However, repeatedly self-handicapping as motivated by threats to the self-concept, may itself threaten self-concept (Schwinger et al., 2022). Recurrent self-handicapping may result in patterns of underperformance which may become incorporated into one's self-concept. This is consistent with reciprocal models of self-concept development, wherein self-concept-motivated behaviours can form patterns which circularly shape self-concept; these will be explored later in this chapter. 

Self-concept is an important guiding system for behaviour. Thus, self-deception through self-defensive strategies can create a dysfunctional self-concept which can have detrimental effects on how behaviour is motivated as well as emotional wellbeing (Greve &amp; Wentura, 2003). Together, this research suggests that self-concept can act both as a motivational resource, and motivational vulnerability. When self-concept is threatened by evidence evaluated to suggest one is not who they think they are, motivation decreases.

== The influence of motivation on self-concept ==
The research and theories have thus far detailed how self-concept directs and motivates behaviour. The following section explores how this relationship works in reverse, illustrating the bidirectional relationship between self-concept and motivation.  

=== How does internalising motivation shape self-concept? ===

* ''Self-determination theory proposes motivation is influenced by needs (competence, autonomy, relatedness) which develop motivations from externally regulated to increasingly intrinsically regulated (Ryan &amp; Deci, 2020).'' 

* ''Identifying with a behaviour shapes motivation to engage with a behaviour; when motivational needs are met, motivations begin to shape self-concept.'' 

* ''Chiu’s (2023) study involved teaching methods supporting competence, autonomy, and relatedness needs. The intervention resulted in increased identification and interest in STEM. Meeting motivational needs increase self-identification with the subject.  ''

* ''Yip et al.'s (2024) study found self-determined motivations positively correlated with group identification.'' 

'''[[Self-determination theory]]''' proposes there are different types of motivation, influenced by three innate needs, competence, autonomy, and relatedness (Ryan &amp; Deci, 2020). Understanding the temporal progression of these needs and motivation types can provide insight into how motivational processes can influence self-concept.  

Ryan and Deci (2020) suggest behavioural regulation exists on a continuum of self-determination, ranging from externally regulated behaviour to increasingly internalised regulation. The motivations proposed on the self-determination continuum, in order of increasingly self-determined motivations are: external, introjected, identified, integrated, and intrinsic regulation. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 2). Extrinsic motivation is the drive to obtain an external reward or avoid a negative result (Morris et al., 2022). The degree of how self-determined one's motivations are depends on the extent to which they have internalised the values, beliefs and perceptions of their behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised (Chiu, 2023). Extrinsic motivation and associated values and behaviours become increasingly integrated with the self and increasingly self-endorsed. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when motivational needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 2, Van Den Broeck et al., 2021).  

'''Table 2'''

Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development.
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
!Type of motivation 
!Ash's circumstances  
!Mechanism 
|-
|External regulation 
|Ash's parents signed her up to swim. After swimming lessons she would receive a lollipop.  
|Non-self-determined drive.  
|-
|Introjected regulation 
|Ash starts to want her parents to think she is dedicated.   
|Partially internalised extrinsic motivation, low self-determination.  
|-
|Identified regulation 
|Swimming becomes important to Ash as she begins to feel competent and can drive herself to training (autonomy).  
|More internalised and self-determined. 
|-
|Integrated regulation 
|Ash starts to identify as a swimmer and with her swim team (relatedness).  
|Fully integrated into self-concept, highly self-determined but performed for valued outcome not enjoyment. 
|-
|Intrinsic motivation 
|After the swimming competition, Ash continues swimming for the joy of it.  
|Fully autonomous motivation. 
|}
Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In this study, 342 Hong Kong high school students completed STEM activities taught using a regular teaching approach (non-self-determination theory) or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness. Self-report measures of perceived teacher support, STEM identity and STEM interest reflected that the needs-supporting teaching approach increased students' autonomy, competence and relatedness. This resulted in increased STEM identity and interest. This supports the developmental pathway of self-determination theory proposed whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. 

Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.  

Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming gradually transformed the activity into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. This theory indicates self-concept and motivation are inherently intertwined.  

=== How does the reciprocal effects model explain the influence of motivation on self-concept? ===

* ''Reciprocal effects model suggests self-concept and achievement reinforce each other (Marsh et al., 2018).  ''

* ''Sewasew et al.'s (2018) study demonstrated mathematics self-concept positively predicted mathematics achievement and vice versa. These effects were moderated by motivational constructs of goal approach theory (Vandewalle et al., 2019).'' 

* ''Garn and Shen’s (2015) study found motivational needs (self-determination theory, Ryan &amp; Deci, 2022) did not predict physical self-concept. Physical self-concept was only an antecedent to exercise motivation.'' 

* ''Sorjonen et al. (2024) reanalysed meta-analytic results (Wu et al., 2021) which had provided evidence for the reciprocal effects model. They found that employing different statistical models allowed results to both support and refute the reciprocal relationship''. 

The '''reciprocal effects model''' proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). Individuals with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how individuals behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was a positive predictor of mathematics achievement and vice versa. The effect of achievement on self-concept was moderated by motivational constructs of goal approach theory (Vandewalle et al., 2019). Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation.  

However, Garn and Shen (2015) using a longitudinal model with 329 participants found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan &amp; Deci, 2022) did not predict physical self-concept at follow ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and may be explained by the timeframe of study; Garn and Shen (2015) suggest initial exercise engagement helps one feel good and motivated, thus immediately influencing self-concept, but as novelty wears off, so does motivation and any effects on self-concept. Another possible explanation for why no reciprocal effect was identified here but there was in Sewasew et al.'s (2018) study of mathematical self-concept is that reciprocal effects may be domain specific. The multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not necessarily generalise to another. 

Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results (Wu et al., 2021) which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that employing different statistical models allowed results to both support and refute the finding that motivation as measured by achievement influenced self-concept. Hübner et al. (2023) similarly support for the reciprocal effects model was inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model.  

Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. 

== Self-concept-based interventions ==
The relationship between self-concept and motivation detailed thus far is useful for establishing motivational interventions. This section explores how self-concept and motivation theory can inform effective interventions to improve motivation and outcomes.

=== Foster domain-specific self-concept ===

* ''Self-concept is multidimensional (Marsh et al., 1992).'' 

* ''This suggests interventions may be more effective if they focus on specific identities, rather than global self-love.''

* ''Genuine achievement opportunities are also important - Vu et al. (2024) found that only enhancing self-concept reduces learning effort, which negatively impacts achievement, consequently impacting self-concept.''

As the multidimensional model posits, self-concept is multidimensional and hierarchical (Marsh et al., 2019). Research supports that behaviour and outcomes are more closely linked to specific domains of self-concept than general self-concepts (Seaton et al., 2014; Sewasew et al., 2018). Therefore, motivational self-concept interventions should target specific behaviours and identities, not global self-concept. However, it is important for interventions to foster ''accurate'', positive self-concepts for specific domains. As Vu et al. (2024) note, only enhancing self-concept may reduce learning effort and thus negatively impact achievement, consequently impacting self-concept. Similarly, findings challenging the reciprocal effects model indicate that self-concept interventions should be combined with genuine opportunities for achievement (Sewasew et al., 2018; Sorjonen et al., 2024). Nevertheless, academic self-concept plays a role, and its motivational properties should not be overlooked. This means interventions should focus on developing accurate positive beliefs about subject-specific capabilities, through achievement opportunities, rather than broadly encouraging feeling good about oneself.

=== Foster identity-based motivation ===

* ''Focus on identity-based motivation, not simply outcome-based motivation because identities drive goal pursuit (Oyserman, 2018; Rogers, 1959).''

* ''Lohbeck et al.'s (2021) study found that improving physical activity and motivation by improving physical self-concept was more effective than only using external motivators or increasing activity.''

* ''Identity-aligned goals make behaviour and effort meaningful, and therefore sustainable.''

As Rogers (1959) and Oyserman (2018) explain, our identities drive goal pursuit; individuals are motivated to behave in ways that align with their current or desired self-concept. This framework indicates interventions should not focus on outcomes as goals, but identities as motivators. For example, Ash may have greater motivation to train for her competition when her goal is identity focused, rather than if her goal is to achieve a specific time. This is supported by Lohbeck et al.'s (2021) study of children's physical self-concept, motivation and activity. They found that improving physical activity and motivation by improving physical self-concept may be more effective than interventions which focus exclusively on external motivators, or increased activity. While these findings cannot definitively establish causation, paired with pertinent theories, results reinforce that idea that behaviour change and outcome goals may not be as effective motivators as self-concept-aligned goals. Together, identity-aligned goals make behaviour and effort meaningful, and therefore sustainable, which is especially important to maintain persistence when progress is slow.  

=== Reduce self-threat ===

* ''Success as a contingency for self-worth can be damaging (Lawrence &amp; Gonzales, 2022; Showers et al., 2015)'' 

* ''This can be detrimental, as research supports self-concept is core to wellbeing: Li et al. (2025) found that self-concept positively predicted wellbeing, mediated by gratitude and prosocial behaviour interventions.'' 

* ''Interventions, through non-success-based practices like gratitude or prosocial habits, should focus on deemphasising success as core to self-worth.'' 

A growing body of research indicates that fostering a coherent sense of self is core to wellbeing (Sirgy, 2021). For example, Li et al. (2025) found that self-concept positively predicted wellbeing, mediated by gratitude and prosocial behaviour interventions. As illustrated, however, by the human tendency to self-handicap to defensively protect self-concept, holding success as a contingency for self-worth can be damaging (Lawrence &amp; Gonzales, 2022; Showers et al., 2015). Wellbeing-focused interventions should assist in deemphasising the extent to which self-worth depends on actual or expected outcomes. Promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure may therefore improve both wellbeing and motivation.

Across academic, physical, and wellbeing domains, effective interventions are unlikely to only target behaviour. Rather, interventions that shape how individuals understand themselves may produce more sustainable and high quality motivation by influencing the self-concepts that guide behaviour.

== Conclusion ==
Self-concept is one's internal perception of oneself. Self-concept acts as a guiding compass that motivates and directs behaviours through processes of congruence, perceived self-competence, and responses to failure. While self-concept can help individuals make sense of and strive toward goals (Oyserman, 2024; Rogers, 1959) evaluations of competence (Vandewalle et al., 2019) and contingencies of self-worth (Showers et al., 2015) can undermine motivation. 

Motivation and subsequent behaviours also form self-concept through internalisation processes when psychological needs are met, as proposed by the self-determination theory (Ryan &amp; Deci, 2020). However, inconsistent support for the reciprocal effects model (Garn &amp; Shen, 2015; Sewasew et al., 2018) emphasises that motivation may not have as great a role in shaping self-concept as once thought, underscoring the complexity of these processes. 

Nevertheless, these processes inform approaches to motivation-improving interventions. Self-concept-based interventions for improving motivation should focus on domain-specific and identity-based motives, which deemphasise contingencies of self-worth. With self-concept and motivation inherently linked, who one believes they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who one becomes. 

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'''Table 1'''

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g.,

The 2 x 2 Johari Window Model Showing Knowing x Self/Other
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
!
!Known to self
!Not known to self
|-
|'''Known to others'''
|Open area
|Blind spot
|-
|'''Not known to others'''
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|Unknown
|}

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|type=&quot;()&quot;}
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== See also ==
* [[wikipedia:Self-concept|Self-concept]] (Wikipedia)
* [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019)
* [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014)
== References ==
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Boldero, J., &amp; Francis, J. (2002). Goals, standards, and the self: Reference values serving different functions. ''Personality &amp; Social Psychology Review (Lawrence Erlbaum Associates), 6''(3), 232–241. https://doi.org/10.1207/S15327957PSPR0603_7  

Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8  

Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126-28.  

Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., &amp; Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237  

Crocker, P. R. E., Sabiston, C. M., Kowalski, K. C., McDonough, M. H., &amp; Kowalski, N. (2006). Longitudinal assessment of the relationship between physical self-concept and health-related behavior and emotion in adolescent girls. ''Journal of applied sport psychology, 18''(3), 185–200. https://doi.org/10.1080/10413200600830257  

Fairlamb, S. (2022). We need to talk about self-esteem: The effect of contingent self-worth on student achievement and well-being. ''Scholarship of Teaching and Learning in Psychology, 8''(1), 45–57. https://doi/10.1037/stl0000205  

Frade, A. S. B. V., &amp; Veiga, F. H. (2017). Student motivation and self-concept: Is there a connection? ''The European Proceedings of Social and Behavioural Sciences, 31'', 203–213. https://doi.org/10.15405/epsbs.2017.10.19  

Garn, A., &amp; Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994  

Greve, W., &amp; Wentura, D. (2003). Immunizing the self: Self-concept stabilization through reality-adaptive self-definitions. ''Personality and Social Psychology Bulletin, 29''(1), 39–50. https://doi.org/10.1177/0146167202238370   

Howard, C., &amp; Dhillon, J. K. (2022). Q-sort as a tool for promoting outstanding leadership. ''Practice, 4''(1), 33–41. https://doi.org/10.1080/25783858.2021.1882264 

Hübner, N., Wagner, W., Zitzmann, S., &amp; Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6  

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Lawrence, J. S., &amp; Gonzales, J. E. (2022). Academically-contingent self-worth: Different dimensions differentially predict future vulnerability. ''Current Psychology 42'', 24947–24961 (2023). https://doi.org/10.1007/s12144-022-03516-x  

Lens, W., &amp; Vansteenkiste, M. (2020). Motivation: About the “why” and “what for” of human behavior. In ''Psychological concepts'' (pp. 249-270). Psychology Press. https://www.taylorfrancis.com/chapters/edit/10.4324/9781003076384-12/motivation-human-behavior-willy-lens-maarten-vansteenkiste  

Li, J., Li, Y., &amp; Ding, Y. (2025). Self-concept promote subjective well-being through gratitude and prosocial behavior during early adolescence? A longitudinal study. ''BMC Psychology, 13''(1), 356. https://doi.org/10.1186/s40359-025-02652-w  

Lietz, R., Whittaker, S., &amp; Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour &amp; Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023  

Lohbeck, A., Von Keitz, P., Hohmann, A., &amp; Daseking, M. (2021). Children's physical self-concept, motivation, and physical performance: Does physical self-concept or motivation play a mediating role? ''Frontiers in Psychology, 12''. https://doi.org/10.3389/fpsyg.2021.669936  

Marsh, H. W., Byrne, B. M., &amp; Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt &amp; R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. 

Marsh, H. W., Martin, A. J., Yeung, A. S., &amp; Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, &amp; D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85-115). Guilford Publications.  

Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., &amp; Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, &amp; S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36-62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004  

Marsh, H. W., &amp; Shavelson, R. (1985). Self-concept: Its multifaceted, hierarchical structure. ''Educational Psychologist, 20''(3), 107–123. https://doi.org/10.1207/s15326985ep2003_1  

Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., &amp; Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611https://doi.org/10.1017/S0033291722001611 

Nurra, C., &amp; Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003  

Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1-11. https://doi.org/10.1002/9781118900772.etrds0171  

Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028  

Phillips, E. L., Raiford, A., &amp; El-Batrawi, S. (1965). The Q sort reevaluated. ''Journal of Consulting Psychology, 29''(5), 422–425. https://doi.org/10.1037/h0022506  

Rhodewalt, F., &amp; Fairfield, M. (1991). Claimed self-handicaps and the self-handicapper: The relation of reduction in intended effort to performance. ''Journal of Research in Personality, 25''(4), 402-417. https://doi/10.1016/0092-6566(91)90030-T  

Rogers, C. R. (1959). A theory of therapy, personality, and interpersonal relationships: As developed in the client-centered framework. In S. Koch (Ed.), ''Psychology: A study of a science'' (Vol. 3, pp. 184-256). McGraw-Hill. 

Ryan, R. M., &amp; Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860  

Ryan, R. M., &amp; Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1-7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2  

Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., &amp; Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706  

Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., &amp; Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232  

Sewasew D., Schroeders U., Schiefer I. M., Weirich S., &amp; Artelt C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003  

Showers, C. J., Ditzfeld, C. P., &amp; Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130  

Siljanovska, L., &amp; Stojcevska, S. (2018). A critical analysis of interpersonal communication in modern times of the concept “Looking Glass Self (1902)” by Charles Horton Cooley. ''Seeu Review, 13''(1), 62-74. https://doi.org/10.2478/seeur-2018-0007  

Sirgy, M. J. (2021). Effects of self-concept on wellbeing. In A. C. Michalos (Ed.), ''The psychology of quality of life'' (pp. 307-320). Springer, Cham. https://doi.org/10.1007/978-3-030-71888-6_14  

Sorjonen, K., Ingre, M., Melin, B., &amp; Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826  

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Zhang, J., Zhang, Y., Song, Y., &amp; Gong, Z. (2016). The different relations of extrinsic, introjected, identified regulation and intrinsic motivation on employees’ performance. ''Management Decision, 54''(10), 2393–2412. https://doi.org/10.1108/md-01-2016-0007  }}{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
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* The most common mistakes include:
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== External links ==
* [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast)
* [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept]  (TEDx Talks)
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== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]]

;Scenario

Ash is a swimmer training for a national competition. Being &quot;a swimmer&quot; is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00am every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle?
{{RoundBoxBottom}}

Your self-concept is your understanding of who you are and how you feel about yourself. This includes a complex system of the attitudes, beliefs and judgements you hold about yourself. Self-concept guides you in answering the question “who am I?” (Wehrle &amp; Fasbender, 2018). But its importance extends beyond answering that question. Self-concept influences how you think, feel and behave. Consider self-concept an internal compass which motivates and guides your choices, relationships and understanding of the world.  

For Ash in the case study, being a swimmer and succeeding in the upcoming swimming competition are core to her self-concept. This guides and motivates her behaviour of swimming early every morning.  

This chapter explores how our understanding of ourselves influences what we’re motivated to do and examines how what we repeatedly do subsequently shapes our self-concept. By understanding how and why we perceive ourselves the way we do, and the impacts of our self-concepts, we can better understand our behaviours, foster positive self-concept and embrace the motivational properties of self-concept.{{RoundBoxTop|theme=3}}
'''Focus questions'''

*What is self-concept? 
*What are the key theories of self-concept? 
*How does self-concept influence motivation? 
*How does motivation influence self-concept?
*How can our understanding of self-concept be used to improve motivation and wellbeing?{{RoundBoxBottom}}

== Understanding self-concept ==
The self is central to humanhood and the scientific understanding of human behaviour. Thus, the question of “what makes me, me?” has been theorised about throughout history and is important to understand. This section details the core theories and properties of self-concept which will be referred to throughout the chapter.  

Self-concept is a cognitive framework which is unique to, but encompasses, many of the popular “self-” constructs – [[self-esteem]] (perceived value of yourself), [[w:Self-efficacy|self-efficacy]] (perceived task capability) and [[w:Self-image|self-image]] (description of yourself). See the linked pages for details on each of these separate constructs. 

==== William James ====
James (1892) proposed the self consists of two parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. This philosophical framework laid the basis for the development of empirical theories of self-concept and catapulted psychological identity research.  

==== The looking-glass self ====
Cooley (1902) framed the individual as a social being, theorising that self-concept is a reflection of the responses and evaluations of individuals within one’s environment. Self-concept is developed through the imagined judgement of others, and individuals become the people others say they are (Siljanovska &amp; Stojcevska, 2018). 

==== Rogers’ self theory ====
Rogers (1959), a humanist psychologist, proposed self-concept is a dynamic, organised system which consists of three parts: the real self (how you see yourself), ideal self (who you would like to be), and self-worth (how much you value yourself). He proposed that individuals behave in accordance with their self-concept, a stepping stone toward the contemporary understanding of how self-beliefs influence motivation (Ismail &amp; Tekke, 2015; Yousefi &amp; Kiani, 2014).  

==== The multidimensional self-concept model ====
Marsh et al. (1992) were critical of the lack of theoretical and methodological rigor supporting studies of self-concept. They built upon previous theories, suggesting self-concept is the perception of oneself shaped by experience and evaluations, but proposed self-concept is organised, multifaceted and hierarchical (Marsh et al., 2019). An individual’s self-concept is comprised of academic, social, emotional and physical dimensions which each divide into further subsets of increasing specificity and are closer related to actual behaviour (see Figure 2). This original model was later refined by developing and analysing the Self Description Questionnaire and produced a model which captures the correlations between each aspect of self-concept (see Figure 2; Marsh &amp; Shavelson, 1985). This hierarchical model predicts behaviour and feelings more accurately than previous general theories (Crocker et al., 2006; Marsh et al., 2019).  [[File:Multidimensional model of self-concept diagram.png|thumb|600x600px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 1992).|center]]
==== Properties of self-concept   ====
Although these theories each approach self-concept differently, several common properties emerge across them. Self-concept appears to be a multifaceted, hierarchical, socially developed, and dynamic system of self-knowledge (Table 1). These properties provide the foundation for understanding the motivational effects of self-concept. 

'''Table 1'''

Summary Table of Self-Concept Properties and Their Significance for Motivation.
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
!Property
!Theory
!Significance for motivation
|-
|'''Multifaceted'''
|James, Rogers, Multidimensional model
|Different aspects of the self motivate different behaviours
|-
|'''Hierarchical  '''
|Multidimensional model
|General and specific domains influence each other and different behaviours
|-
|'''Socially developed'''
|The looking glass self
|Social feedback influences which behaviours are pursued  
|-
|'''Dynamic'''
|Rogers, Multidimensional model
|Self-concept changes with experience
|-
|'''Evaluative'''
|Rogers
|Perceived self-worth affects responses to success/failure
|-
|'''Descriptive  '''
|Rogers, Multidimensional model
|Identity beliefs influence goals and expectations
|}

== The influence of self-concept on motivation ==
'''''[Author's note: I have begun drafting out the chapter in full-sentence form. I am keeping my planned key points for each''''' '''''of the following sections in italics.]''''' 

* ''A growing body of research suggests self-concept has important motivational properties.'' 

* ''Motivation is an internal process which initiates, directs and sustains behaviour (Lens &amp; Vansteenkiste, 2020).''

There is a growing body of research suggesting self-concept has important motivational properties. Motivation is an internal process which initiates, directs and sustains behaviour (Lens &amp; Vansteenkiste, 2020). Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation are both central guiding forces of behaviour and navigating the environment. This section explores how self-concept influences motivation.  

=== How does self-concept direct goals? ===

* ''Rogers’ (1959) congruence theory explains people are driven to behave in ways to conform to their real and ideal selves.''

* ''Empirical testing of congruence produces mixed results (Boldero &amp; Francis, 2002; Howard &amp; Dhillon, 2022; Phillips et al., 1965).'' 

* ''Oyserman’s (2024) identity-based motivation theory provides a more flexible account of how self-concept influences behaviour in everyday situations, suggesting identities are contextually salient.  ''

* ''When behaviour is congruent with one’s identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). Incongruent actions that have the same challenges are interpreted as pointless. Identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024).'' 

* ''Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, increased when children were guided to connect with their future selves.'' 

* ''Self-concept directs behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted.'' 

Individuals pursue and evaluate goals in ways that align with their self-concept. This is driven by the human tendency to strive for alignment between self-concept and behaviour.  

'''Rogers’ (1959) theory''' suggests the self is comprised of the real self and the ideal self. The alignment of these is known as congruence, a state where one has the capacity to pursue self-improving goals. When the ideal and real self do not align (incongruence), an individual may experience poorer mental health and act defensively. This incongruence generates a psychological tension which acts as a motivating force, pushing the individual to either modify their behaviour to align with their self-concept, or modify their self-concept (Stephen, 2023).  

Supporting this concept, later research has similarly suggested that ideal standards function as reference values individuals use to evaluate their current behaviour and direct goal pursuit (Boldero &amp; Francis, 2002). Howard and Dhillon (2022) also found using Rogers’ Q-sort, a measure of self-concept and congruency, can be an effective method of reflection to help individuals consciously choose goals that help achieve a congruent, ideal self.  

However, direct support for the foundational concept of congruence is sparse (Phillips et al., 1965). Rogers’ Q-sort was used to empirically test the concept of congruence. Phillips et al. (1965) found that incongruent participants, as indicated by Q-sort results, did not become more congruent over a 6-month measurement period, thus indicating a lack of movement motivated by incongruency. One explanation is that Rogers conceptualised self-concept as relatively stable. But this framework has not been fully discounted. In response to the poor empirical support for congruence, contemporary research tends to adopt a dynamic and context dependent approach to self-concept. Oyserman’s (2024) identity-based motivation theory, for example considers self-concept as continually constructed dynamically and contextually. This approach may therefore provide a more flexible and realistic account of how self-concept influences behaviour in everyday situations. 

'''[[wikipedia:Identity_based_motivation|Identity-based motivation theory]]''' suggests situations influence which aspects of our multidimensional self-concept are active – the most salient identity on one’s mind - and thus influence behaviour. Like Rogers’ theory, it suggests individuals behave and interpret situations in ways that align with their identity. However, Oyserman (2015) expands Roger’s simplistic differentiation between real and ideal selves. The theory suggests a temporal flexibility of self-concept. A future self can be an active identity and thus individuals may act congruently to future selves. This may explain why future identities motivate present behaviour. Unlike Rogers' relatively stable model of the ideal self, identity-based motivation theory suggests future identities can become situationally salient and influence immediate behavioural decisions. 

The theory suggests that when behaviour is congruent with one’s identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). Incongruent actions that have the same challenges are interpreted as pointless. Identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). 

But why is importance motivating? Identity-relevant goals are motivating because they provide information about who one is and who one wants to become. This makes difficulty meaningful rather than discouraging (Oyserman, 2024).
{{RoundBoxTop|theme=3}}'''Case study comprehension check:'''
Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}}

Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children and subsequent achievement increased when children were guided to connect with their future selves. In two separate studies, students were primed to think of their adult-self as near or far. In the first study, those in the adult-self is near group performed significantly better on a geometry task than the control and far group. The second study used a longer-term measure of final grades. Students in the adult-self is near group again performed significantly better than the far group. This longer-term effect of connecting to one’s future self lasted six months. These findings indicate that identifying with one’s future/ideal self can motivate goal attainment. Self-concept is malleable and moderates goal pursuit based on how one perceives themselves at a given time.  

Together, these theories suggest self-concept influences motivation through two complementary processes. Individuals are motivated to reduce discrepancies between their real and ideal selves. Additionally, individuals are motivated to pursue goals that are congruent with situationally active identities. Thus, self-concept directs behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted.  

=== What is the role of perceived competence in the effect of self-concept on motivation? ===

* ''Evaluative beliefs about competence are part of self-concept. Individuals engage with goals they believe they are more capable of achieving (Marsh et al., 2019).'' 

* ''Seaton et al.'s (2014) study found prior mathematics self-concept was a significant positive predictor of mastery-approach and a positive, but weaker predictor of performance-approach goal orientations.'' 

* ''Goal orientation theory: mastery goal approaches involve being motivated to learn, whereas performance goal approaches are motivated by wanting to demonstrate competence (Vandewalle et al., 2019).'' 

* ''Perceived competence provides a motivational framework for how people engage with goals.'' 

Self-concept incorporates evaluative beliefs about competence which influence how individuals approach goals. Individuals engage with goals they believe they are more capable of achieving (Marsh et al., 2019). For example, Seaton et al.'s (2014) longitudinal study of 2786 Australian students' found prior mathematics self-concept was a significant positive predictor of mastery-approach and a positive, but weaker predictor of performance-approach goal orientations. According to '''[[wikipedia:Goal_orientation|goal orientation theory]]''', a mastery goal approach involves being motivated to learn, whereas a performance goal approach is motivated by wanting to demonstrate competence (Vandewalle et al., 2019). These results, therefore, suggest that domain-specific self-concepts provide a motivational foundation for how students engage with learning tasks. When students perceive themselves as competent, they are more likely to be motivated to learn, rather than merely demonstrate their competence to others.  

Frade and Viega's (2017) study of navy trainees corroborated these results with non-domain specific measures and a Portuguese adult population, indicating results are generalisable beyond Seaton et al.'s (2014) Australian high school participants and the domain of mathematics. However, Seaton et al.'s (2014) study did not incorporate avoidance (goals motivated to prevent failure) orientations of mastery and performance goals in their analyses, limiting the deductive power of these findings and possibly understating the strength of the identified correlation. Nevertheless, results support that perceived self-competence influences goal engagement.  

=== How can self-concept undermine motivation? ===

* ''Self-worth contingencies refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). Holding contingent beliefs about one's value can undermine motivation.'' 

* ''Highly contingent self-worth means failing threatens individuals’ identity.'' 

* ''Lietz et al. (2026) found that high contingent self-worth correlated with participants' negative and severe emotional responses to failure. Participants attributed failure to themselves.''

* ''These finding are consistent with attribution theory (Weiner, 1985): causal thinking processes where people attribute outcomes to stable, uncontrollable factors, results in low motivation. Self-attributions of failure can undermine motivation.'' 

* ''Contingent self-worth is temporary (Fairlamb, 2020; Lawrence &amp; Gonzales, 2022). This can be explained by regulation types detailed in self-determination theory (Ryan &amp; Deci, 2020).'' 

* ''Self-handicapping: creating obstacles to your own goal so that failure cannot be self-attributed (Török et al., 2018).'' 

* ''Rhodewalt and Fairfield (1991) found that participants who scored highly on a measure of self-handicapping tendency and reported intentions to withhold effort in a test subsequently performed worse than others.'' 

* ''Self-concept can act as a motivational vulnerability as well as a motivational resource.''

The theories and research discussed so far optimistically discuss how self-concept motivates one to meet their ideal self. However, there are circumstances under which self-concept undermines motivation. Contingent self-evaluations and self-handicapping help us understand how self-concept can undermine motivation. 

'''[[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]]''' refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). Holding contingent beliefs about one's value can undermine motivation. For individuals who have highly contingent self-worth, achieving goals is closely tied to their self-concept, meaning failure is threatening to one's self-concept. Accordingly, contingent self-worth motivates pursuit of success for domains that are important for one's self-worth. This is consistent with Oyserman and Rogers' theories that individuals tend to set self-consistent goals, and that these goals are highly valued and thus highly motivated. 

Lietz et al. (2026) corroborates that this motivating guide provided by self-concept resulted in participants' negative and severe emotional responses to failure. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept, which resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with '''[[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]]''' (Weiner, 1985), which posits that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). 

Presenting another pitfall of self-concept driven motivation, Fairlamb (2020) suggests contingent self-worth motivation is temporary. This temporary nature is illustrated by Lawrence and Gonzales' (2022) results from 466 university students, wherein self-report measures of motivation and self-worth showed a correlation between academically contingent self-worth and self-worth boosting goals such as studying to graduate, rather than studying for the joy of it. They also showed a correlation between academically contingent self-worth and amotivation, a lack of purpose and drive.  

van der Kaap-Deeder et al. (2016) explains the motivational boost from contingent self-worth is temporary because it is driven by a 'need to do well' motivation (introjected regulation), rather than intrinsic regulation (internal drive). These different motivation types are proposed by '''[[self-determination theory]]''', wherein intrinsic motivation provides a more sustainable motivational force, as the behaviour is powered by its inherent satisfaction (Zhang et al., 2016). Introjected motivation, however, is driven by internal guilt or pressure, resulting in stress and a feeling of a lack of autonomy (Taris et al., 2020). These uncomfortable affective experiences, paired with any perceived risk of failure causes motivation from contingent self-worth to decrease, and maladaptive behaviours emerge (van der Kaap-Deeder et al., 2016). 

One such response to the threat of failure is self-handicapping. '''[[w:Self-handicapping|Self-handicapping]]''' is the act of creating obstacles to one's own goal so that failure may not be self-attributed (Török et al., 2018). Creating a handicap generates an equally plausible and external explanation for one's outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, reflecting the motivating quality of self-concept and indicating self-concept can hinder goal pursuit (Coudevylle et al., 2020). 

{{RoundBoxTop|theme=3}}'''Case study comprehension check:'''
Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence.  {{RoundBoxBottom}}

Schwinger et al. (2022) found a medium negative correlation between general self-esteem and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. Self-esteem mirrors the evaluative property of self-concept; fear of failure reflects concern about the implications of failure for one's self-concept. This highlights self-concept as an antecedent to self-handicapping and related motivational and performance outcomes. In this study, general, rather than domain-specific measures were used. As the multidimensional theory of self-concept suggests, the more specific the self-concept facet, the more closely related to actual behaviour it is. Accordingly, future research should incorporate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation.

Self-handicapping demonstrates that motivation is not always directed towards achieving valued outcomes. Under some circumstances, protecting self-worth becomes more important than succeeding at the task itself. To illustrate the behavioural effects of self-handicapping, Rhodewalt and Fairfield (1991) found that participants who scored highly on a measure of self-handicapping tendency and reported intentions to withhold effort in a test subsequently performed worse than others. This reflects the influence of self-concept on motivation; preserving one’s image undermines motivation to succeed due to the threat of failure and what it means for one’s self concept. Whereas Rogers (1959) and Oyserman (2015) suggest people are motivated to pursue identity-congruent goals, self-handicapping highlights a potential downside of identity-relevant goals. When success becomes central to the self, individuals may become motivated to protect their self-concept rather than maximise performance. 

However, repeatedly self-handicapping as motivated by threats to the self-concept, may itself threaten self-concept (Schwinger et al., 2022). Recurrent self-handicapping may result in patterns of underperformance which may become incorporated into one's self-concept. This is consistent with reciprocal models of self-concept development, wherein self-concept-motivated behaviours can form patterns which circularly shape self-concept; these will be explored later in this chapter. 

Self-concept is an important guiding system for behaviour. Thus, self-deception through self-defensive strategies can create a dysfunctional self-concept which can have detrimental effects on how behaviour is motivated as well as emotional wellbeing (Greve &amp; Wentura, 2003). Together, this research suggests that self-concept can act both as a motivational resource, and motivational vulnerability. When self-concept is threatened by evidence evaluated to suggest one is not who they think they are, motivation decreases.

== The influence of motivation on self-concept ==
The research and theories have thus far detailed how self-concept directs and motivates behaviour. The following section explores how this relationship works in reverse, illustrating the bidirectional relationship between self-concept and motivation.  

=== How does internalising motivation shape self-concept? ===

* ''Self-determination theory proposes motivation is influenced by needs (competence, autonomy, relatedness) which develop motivations from externally regulated to increasingly intrinsically regulated (Ryan &amp; Deci, 2020).'' 

* ''Identifying with a behaviour shapes motivation to engage with a behaviour; when motivational needs are met, motivations begin to shape self-concept.'' 

* ''Chiu’s (2023) study involved teaching methods supporting competence, autonomy, and relatedness needs. The intervention resulted in increased identification and interest in STEM. Meeting motivational needs increase self-identification with the subject.  ''

* ''Yip et al.'s (2024) study found self-determined motivations positively correlated with group identification.'' 

'''[[Self-determination theory]]''' proposes there are different types of motivation, influenced by three innate needs, competence, autonomy, and relatedness (Ryan &amp; Deci, 2020). Understanding the temporal progression of these needs and motivation types can provide insight into how motivational processes can influence self-concept.  

Ryan and Deci (2020) suggest behavioural regulation exists on a continuum of self-determination, ranging from externally regulated behaviour to increasingly internalised regulation. The motivations proposed on the self-determination continuum, in order of increasingly self-determined motivations are: external, introjected, identified, integrated, and intrinsic regulation. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 2). Extrinsic motivation is the drive to obtain an external reward or avoid a negative result (Morris et al., 2022). The degree of how self-determined one's motivations are depends on the extent to which they have internalised the values, beliefs and perceptions of their behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised (Chiu, 2023). Extrinsic motivation and associated values and behaviours become increasingly integrated with the self and increasingly self-endorsed. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when motivational needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 2, Van Den Broeck et al., 2021).  

'''Table 2'''

Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development.
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
!Type of motivation 
!Ash's circumstances  
!Mechanism 
|-
|External regulation 
|Ash's parents signed her up to swim. After swimming lessons she would receive a lollipop.  
|Non-self-determined drive.  
|-
|Introjected regulation 
|Ash starts to want her parents to think she is dedicated.   
|Partially internalised extrinsic motivation, low self-determination.  
|-
|Identified regulation 
|Swimming becomes important to Ash as she begins to feel competent and can drive herself to training (autonomy).  
|More internalised and self-determined. 
|-
|Integrated regulation 
|Ash starts to identify as a swimmer and with her swim team (relatedness).  
|Fully integrated into self-concept, highly self-determined but performed for valued outcome not enjoyment. 
|-
|Intrinsic motivation 
|After the swimming competition, Ash continues swimming for the joy of it.  
|Fully autonomous motivation. 
|}
Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In this study, 342 Hong Kong high school students completed STEM activities taught using a regular teaching approach (non-self-determination theory) or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness. Self-report measures of perceived teacher support, STEM identity and STEM interest reflected that the needs-supporting teaching approach increased students' autonomy, competence and relatedness. This resulted in increased STEM identity and interest. This supports the developmental pathway of self-determination theory proposed whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. 

Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.  

Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming gradually transformed the activity into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. This theory indicates self-concept and motivation are inherently intertwined.  

=== How does the reciprocal effects model explain the influence of motivation on self-concept? ===

* ''Reciprocal effects model suggests self-concept and achievement reinforce each other (Marsh et al., 2018).  ''

* ''Sewasew et al.'s (2018) study demonstrated mathematics self-concept positively predicted mathematics achievement and vice versa. These effects were moderated by motivational constructs of goal approach theory (Vandewalle et al., 2019).'' 

* ''Garn and Shen’s (2015) study found motivational needs (self-determination theory, Ryan &amp; Deci, 2022) did not predict physical self-concept. Physical self-concept was only an antecedent to exercise motivation.'' 

* ''Sorjonen et al. (2024) reanalysed meta-analytic results (Wu et al., 2021) which had provided evidence for the reciprocal effects model. They found that employing different statistical models allowed results to both support and refute the reciprocal relationship''. 

The '''reciprocal effects model''' proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). Individuals with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how individuals behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was a positive predictor of mathematics achievement and vice versa. The effect of achievement on self-concept was moderated by motivational constructs of goal approach theory (Vandewalle et al., 2019). Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation.  

However, Garn and Shen (2015) using a longitudinal model with 329 participants found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan &amp; Deci, 2022) did not predict physical self-concept at follow ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and may be explained by the timeframe of study; Garn and Shen (2015) suggest initial exercise engagement helps one feel good and motivated, thus immediately influencing self-concept, but as novelty wears off, so does motivation and any effects on self-concept. Another possible explanation for why no reciprocal effect was identified here but there was in Sewasew et al.'s (2018) study of mathematical self-concept is that reciprocal effects may be domain specific. The multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not necessarily generalise to another. 

Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results (Wu et al., 2021) which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that employing different statistical models allowed results to both support and refute the finding that motivation as measured by achievement influenced self-concept. Hübner et al. (2023) similarly support for the reciprocal effects model was inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model.  

Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. 

== Self-concept-based interventions ==
The relationship between self-concept and motivation detailed thus far is useful for establishing motivational interventions. This section explores how self-concept and motivation theory can inform effective interventions to improve motivation and outcomes.

=== Foster domain-specific self-concept ===

* ''Self-concept is multidimensional (Marsh et al., 1992).'' 

* ''This suggests interventions may be more effective if they focus on specific identities, rather than global self-love.''

* ''Genuine achievement opportunities are also important - Vu et al. (2024) found that only enhancing self-concept reduces learning effort, which negatively impacts achievement, consequently impacting self-concept.''

As the multidimensional model posits, self-concept is multidimensional and hierarchical (Marsh et al., 2019). Research supports that behaviour and outcomes are more closely linked to specific domains of self-concept than general self-concepts (Seaton et al., 2014; Sewasew et al., 2018). Therefore, motivational self-concept interventions should target specific behaviours and identities, not global self-concept. However, it is important for interventions to foster ''accurate'', positive self-concepts for specific domains. As Vu et al. (2024) note, only enhancing self-concept may reduce learning effort and thus negatively impact achievement, consequently impacting self-concept. Similarly, findings challenging the reciprocal effects model indicate that self-concept interventions should be combined with genuine opportunities for achievement (Sewasew et al., 2018; Sorjonen et al., 2024). Nevertheless, academic self-concept plays a role, and its motivational properties should not be overlooked. This means interventions should focus on developing accurate positive beliefs about subject-specific capabilities, through achievement opportunities, rather than broadly encouraging feeling good about oneself.

=== Foster identity-based motivation ===

* ''Focus on identity-based motivation, not simply outcome-based motivation because identities drive goal pursuit (Oyserman, 2018; Rogers, 1959).''

* ''Lohbeck et al.'s (2021) study found that improving physical activity and motivation by improving physical self-concept was more effective than only using external motivators or increasing activity.''

* ''Identity-aligned goals make behaviour and effort meaningful, and therefore sustainable.''

As Rogers (1959) and Oyserman (2018) explain, our identities drive goal pursuit; individuals are motivated to behave in ways that align with their current or desired self-concept. This framework indicates interventions should not focus on outcomes as goals, but identities as motivators. For example, Ash may have greater motivation to train for her competition when her goal is identity focused, rather than if her goal is to achieve a specific time. This is supported by Lohbeck et al.'s (2021) study of children's physical self-concept, motivation and activity. They found that improving physical activity and motivation by improving physical self-concept may be more effective than interventions which focus exclusively on external motivators, or increased activity. While these findings cannot definitively establish causation, paired with pertinent theories, results reinforce that idea that behaviour change and outcome goals may not be as effective motivators as self-concept-aligned goals. Together, identity-aligned goals make behaviour and effort meaningful, and therefore sustainable, which is especially important to maintain persistence when progress is slow.  

=== Reduce self-threat ===

* ''Success as a contingency for self-worth can be damaging (Lawrence &amp; Gonzales, 2022; Showers et al., 2015)'' 

* ''This can be detrimental, as research supports self-concept is core to wellbeing: Li et al. (2025) found that self-concept positively predicted wellbeing, mediated by gratitude and prosocial behaviour interventions.'' 

* ''Interventions, through non-success-based practices like gratitude or prosocial habits, should focus on deemphasising success as core to self-worth.'' 

A growing body of research indicates that fostering a coherent sense of self is core to wellbeing (Sirgy, 2021). For example, Li et al. (2025) found that self-concept positively predicted wellbeing, mediated by gratitude and prosocial behaviour interventions. As illustrated, however, by the human tendency to self-handicap to defensively protect self-concept, holding success as a contingency for self-worth can be damaging (Lawrence &amp; Gonzales, 2022; Showers et al., 2015). Wellbeing-focused interventions should assist in deemphasising the extent to which self-worth depends on actual or expected outcomes. Promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure may therefore improve both wellbeing and motivation.

Across academic, physical, and wellbeing domains, effective interventions are unlikely to only target behaviour. Rather, interventions that shape how individuals understand themselves may produce more sustainable and high quality motivation by influencing the self-concepts that guide behaviour.

== Conclusion ==
Self-concept is one's internal perception of oneself. Self-concept acts as a guiding compass that motivates and directs behaviours through processes of congruence, perceived self-competence, and responses to failure. While self-concept can help individuals make sense of and strive toward goals (Oyserman, 2024; Rogers, 1959) evaluations of competence (Vandewalle et al., 2019) and contingencies of self-worth (Showers et al., 2015) can undermine motivation. 

Motivation and subsequent behaviours also form self-concept through internalisation processes when psychological needs are met, as proposed by the self-determination theory (Ryan &amp; Deci, 2020). However, inconsistent support for the reciprocal effects model (Garn &amp; Shen, 2015; Sewasew et al., 2018) emphasises that motivation may not have as great a role in shaping self-concept as once thought, underscoring the complexity of these processes. 

Nevertheless, these processes inform approaches to motivation-improving interventions. Self-concept-based interventions for improving motivation should focus on domain-specific and identity-based motives, which deemphasise contingencies of self-worth. With self-concept and motivation inherently linked, who one believes they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who one becomes. 

== Learning features ==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}

; Scenarios

* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[Motivation and emotion/Book/2026/Self-concept and motivation#Feature boxes|feature boxes]]

{{anchor|Feature box}}

; Feature boxes

* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[Motivation and emotion/Book/2026/Self-concept and motivation#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

; Embedded links

* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[wikipedia:Dreams|dreams]]) of dreams was provided by [[wikipedia:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}

; Tables

* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1'''

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g.,

The 2 x 2 Johari Window Model Showing Knowing x Self/Other
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
!
!Known to self
!Not known to self
|-
|'''Known to others'''
|Open area
|Blind spot
|-
|'''Not known to others'''
|Hidden area
|Unknown
|}

; Quizzes

* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:&lt;quiz display=&quot;simple&quot;&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

== See also ==
* [[wikipedia:Self-concept|Self-concept]] (Wikipedia)
* [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019)
* [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014)
== References ==
{{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., &amp; Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177   

Boldero, J., &amp; Francis, J. (2002). Goals, standards, and the self: Reference values serving different functions. ''Personality &amp; Social Psychology Review (Lawrence Erlbaum Associates), 6''(3), 232–241. https://doi.org/10.1207/S15327957PSPR0603_7  

Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8  

Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126-28.  

Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., &amp; Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237  

Crocker, P. R. E., Sabiston, C. M., Kowalski, K. C., McDonough, M. H., &amp; Kowalski, N. (2006). Longitudinal assessment of the relationship between physical self-concept and health-related behavior and emotion in adolescent girls. ''Journal of applied sport psychology, 18''(3), 185–200. https://doi.org/10.1080/10413200600830257  

Fairlamb, S. (2022). We need to talk about self-esteem: The effect of contingent self-worth on student achievement and well-being. ''Scholarship of Teaching and Learning in Psychology, 8''(1), 45–57. https://doi/10.1037/stl0000205  

Frade, A. S. B. V., &amp; Veiga, F. H. (2017). Student motivation and self-concept: Is there a connection? ''The European Proceedings of Social and Behavioural Sciences, 31'', 203–213. https://doi.org/10.15405/epsbs.2017.10.19  

Garn, A., &amp; Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994  

Greve, W., &amp; Wentura, D. (2003). Immunizing the self: Self-concept stabilization through reality-adaptive self-definitions. ''Personality and Social Psychology Bulletin, 29''(1), 39–50. https://doi.org/10.1177/0146167202238370   

Howard, C., &amp; Dhillon, J. K. (2022). Q-sort as a tool for promoting outstanding leadership. ''Practice, 4''(1), 33–41. https://doi.org/10.1080/25783858.2021.1882264 

Hübner, N., Wagner, W., Zitzmann, S., &amp; Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6  

Ismail, N. A. H., &amp; Tekke, M. (2015). Rediscovering Rogers’s self theory and personality. ''Journal of Educational, Health and Community Psychology, 4''(3), 28-36.  

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Jansen, M., Scherer, R., &amp; Schroeders, U. (2015). Students' self-concept and self-efficacy in the sciences: Differential relations to antecedents and educational outcomes. ''Contemporary Educational Psychology, 41'', 13-24. https://doi/10.1016/j.cedpsych.2014.11.002  

Lawrence, J. S., &amp; Gonzales, J. E. (2022). Academically-contingent self-worth: Different dimensions differentially predict future vulnerability. ''Current Psychology 42'', 24947–24961 (2023). https://doi.org/10.1007/s12144-022-03516-x  

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Lietz, R., Whittaker, S., &amp; Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour &amp; Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023  

Lohbeck, A., Von Keitz, P., Hohmann, A., &amp; Daseking, M. (2021). Children's physical self-concept, motivation, and physical performance: Does physical self-concept or motivation play a mediating role? ''Frontiers in Psychology, 12''. https://doi.org/10.3389/fpsyg.2021.669936  

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Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., &amp; Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, &amp; S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36-62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004  

Marsh, H. W., &amp; Shavelson, R. (1985). Self-concept: Its multifaceted, hierarchical structure. ''Educational Psychologist, 20''(3), 107–123. https://doi.org/10.1207/s15326985ep2003_1  

Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., &amp; Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611https://doi.org/10.1017/S0033291722001611 

Nurra, C., &amp; Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003  

Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1-11. https://doi.org/10.1002/9781118900772.etrds0171  

Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028  

Phillips, E. L., Raiford, A., &amp; El-Batrawi, S. (1965). The Q sort reevaluated. ''Journal of Consulting Psychology, 29''(5), 422–425. https://doi.org/10.1037/h0022506  

Rhodewalt, F., &amp; Fairfield, M. (1991). Claimed self-handicaps and the self-handicapper: The relation of reduction in intended effort to performance. ''Journal of Research in Personality, 25''(4), 402-417. https://doi/10.1016/0092-6566(91)90030-T  

Rogers, C. R. (1959). A theory of therapy, personality, and interpersonal relationships: As developed in the client-centered framework. In S. Koch (Ed.), ''Psychology: A study of a science'' (Vol. 3, pp. 184-256). McGraw-Hill. 

Ryan, R. M., &amp; Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860  

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Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., &amp; Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706  

Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., &amp; Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232  

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* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
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== External links ==
* [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast)
* [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept]  (TEDx Talks)
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      <text bytes="10056" sha1="t2n5hkbcujoqf3zjroqntzdaouz25xe" xml:space="preserve">{{title|Melatonin and Seasonal Mood Changes:&lt;br&gt;What role does melatonin play in seasonal mood changes?}}
__TOC__
[[File:Melatonin.svg|center|thumb|350x350px|Figure 1. Melatonin Chemical Structure]]
==Overview==

{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Case Study: Seasonal Levels of Melatonin near the Arctic Circle

Through explorations of the Arctic circle and the effects on mood and cognitive performance has on explorers, a study was conducted on why this would be the case. 

Circumpolar environments have been known for its changes in day night cycles as well as its temperature. 

Explorers within these research stations have reported increases in depressive symptoms. 

There was a hypothesis on being a link between melatonin production and the effects of the Arctic's seasonal characteristics. Especially the link of the light intensity and its suppression in producing melatonin. 


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The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.
*

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between weather and criminal behaviour? (closed-ended)
* What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Melatonin==
[[w:Melatonin|Melatonin]] is a hormone, best know through its function of regulating circadian rhythms. 

* Indolamines, a family of neurotransmitters, that share a common molecular structure
* Produced in the Pineal Gland 
** For vertabres melatonin is produced in the dark
** Melatonin is a key signaling molecule that connects changes in external light conditions with changes
*** After the retina receives a light signal, it transmits the signal to the pineal gland which then transforms the light signal into melatonin.
*Circulars through the blood stream 
*Various tissues and oragans can secrete melatonin 
**but only the reinal tissue of the pineal glad and eyes show periodic secretory activity
**linking the biosysynthesis of melatonin shows that circidian rhythm and seasonal characteristics associated with external light conditions
**

=== Functions ===

* Sleep (regulates circadian changes)
**
*linked with light
*proctective effects on the nervous system
*Broad spectrum and antibacterial and immunoregularory functions 
*Even looking into melatonin's relations with fertility and reproduction 

=== Melatonin Synthesis and secretion ===
[[File:Melatonin mechanism.jpg|thumb|Figure 2. Mechanism of melatonin biosynthesis]]

* tryptohan excreted from the small instestine into the blood circulation and actively taken up by pineal cells
* AANAT converts the tryptohan hydroxlase into ASMT 
* catalyzes into melatonin
* cAMP-catalyzed proteins promotates formation of a complex 
* reduces the K (m) of serotonin, enhancing melatonin production
* Melatonin is released into the blood and spinal fluid
* blinds into plasma proteins
* Distructed throughout most tissues 

[[File:Melatonin production in 24 hour cycle.jpg|center|thumb|431x431px|Figure 3. Melatonin production in 24 hour cycle]]

=== Melatonin Fluctuations/Imbalance ===

* Melatonin release cycles over time 
* Temperature
* Circandian cycle and daylight changes

&lt;quiz display=&quot;simple&quot;&gt;
{Melatonin is produced by the?
|type=&quot;()&quot;}
+ Pineal Gland
- Cerebral Cortex
- Brain Stem 
- Retina 
 &lt;/quiz&gt;

== Seasonal Mood Disorders ==
Seasonal Affective Disorder (SAD)

* Part of the depression and bipolar family
*

What are they 

Why are they 

== Studies ==

* Artic Study on melatonin and light 
* Effect on controlled-release melatonin on sleep quality, mood and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour (2003)
** Double-blind trial
*** Group A Melatonin 
*** Group B placebo 
** Connected with the photoperiod not the weather-related aspect of the seasons 

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
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;Quizzes
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+ True
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==Conclusion==
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* What are the answers to the focus questions?
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}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[w:Melatonin|Melatonin]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

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}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Megha, K.B., Arathi, A., Shikha, S. et al. Significance of Melatonin in the Regulation of Circadian Rhythms and Disease Management. Mol Neurobiol 61, 5541–5571 (2024). https://doi.org/10.1007/s12035-024-03915-0

Meliska, C. J., Martínez, L. F., López, A. M., Sorenson, D. L., Nowakowski, S., Kripke, D. F., Elliott, J., &amp; Parry, B. L. (2013). Antepartum depression severity is increased during seasonally longer nights: relationship to melatonin and cortisol timing and quantity. Chronobiology international, 30(9), 1160–1173. https://doi.org/10.3109/07420528.2013.808652

}}

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* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
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** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
* Antepartum Depression Severity is Increased During Seasonally Longer Nights: Relationship to Melatonin and Cortisol Timing and Quantity
* Effect of controlled-release melatonin on sleep quality, mood, and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour
* Seasonal levels of melatonin, thyroid hormones, mood, and cognition near the Arctic Circle 
* Melatonin as a treatment for mood disorders: a systematic review
* Melatonin and agomelatine for preventing seasonal affective disorder
* Seasonal rhythms in mental health: a narrative scoping review of neurobiological mechanisms and therapeutic interventions   
* Melatonin Biosynthesis and regulation in reproduction 
* Seasonal changes in mood and behavior: A cluster analytic approach https://www.sciencedirect.com/science/article/abs/pii/0165178189900498 

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__TOC__
[[File:Melatonin.svg|center|thumb|350x350px|Figure 1. Melatonin Chemical Structure]]
==Overview==

{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Case Study: Seasonal Levels of Melatonin near the Arctic Circle

Through explorations of the Arctic circle and the effects on mood and cognitive performance has on explorers, a study was conducted on why this would be the case. 

Circumpolar environments have been known for its changes in day night cycles as well as its temperature. 

Explorers within these research stations have reported increases in depressive symptoms. 

There was a hypothesis on being a link between melatonin production and the effects of the Arctic's seasonal characteristics. Especially the link of the light intensity and its suppression in producing melatonin. 


{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.
*

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* How is Melatonin produced?
* What seasonal effects have been shown on mood?
* Does irregular melatonin production have an impact on Seasonal Mood Disorders and if so why?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between weather and criminal behaviour? (closed-ended)
* What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Melatonin==
[[w:Melatonin|Melatonin]] is a hormone, best know through its function of regulating circadian rhythms. (Megha, et al, 2024)

* Indolamines, a family of neurotransmitters, that share a common molecular structure
* Produced in the Pineal Gland 
** For vertabres melatonin is produced in the dark
** Melatonin is a key signaling molecule that connects changes in external light conditions with changes
*** After the retina receives a light signal, it transmits the signal to the pineal gland which then transforms the light signal into melatonin.
*Circulars through the blood stream 
*Various tissues and oragans can secrete melatonin 
**but only the reinal tissue of the pineal glad and eyes show periodic secretory activity
**linking the biosysynthesis of melatonin shows that circidian rhythm and seasonal characteristics associated with external light conditions
** Meliska et al (2013)

=== Functions ===

* Sleep (regulates circadian changes)
**
*linked with light
*proctective effects on the nervous system
*Broad spectrum and antibacterial and immunoregularory functions 
*Even looking into melatonin's relations with fertility and reproduction 

=== Melatonin Synthesis and secretion ===
[[File:Melatonin mechanism.jpg|thumb|Figure 2. Mechanism of melatonin biosynthesis]]

* tryptohan excreted from the small instestine into the blood circulation and actively taken up by pineal cells
* AANAT converts the tryptohan hydroxlase into ASMT 
* catalyzes into melatonin
* cAMP-catalyzed proteins promotates formation of a complex 
* reduces the K (m) of serotonin, enhancing melatonin production
* Melatonin is released into the blood and spinal fluid
* blinds into plasma proteins
* Distructed throughout most tissues 

[[File:Melatonin production in 24 hour cycle.jpg|center|thumb|431x431px|Figure 3. Melatonin production in 24 hour cycle]]

=== Melatonin Fluctuations/Imbalance ===

* Melatonin release cycles over time 
* Temperature
* Circandian cycle and daylight changes

&lt;quiz display=&quot;simple&quot;&gt;
{Melatonin is produced by the?
|type=&quot;()&quot;}
+ Pineal Gland
- Cerebral Cortex
- Brain Stem 
- Retina 
 &lt;/quiz&gt;

== Seasonal Mood Disorders ==
Seasonal Affective Disorder (SAD)

* Part of the depression and bipolar family
*

What are they 

Why are they 

== Studies ==

* Artic Study on melatonin and light 
* Effect on controlled-release melatonin on sleep quality, mood and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour (2003)
** Double-blind trial
*** Group A Melatonin 
*** Group B placebo 
** Connected with the photoperiod not the weather-related aspect of the seasons 

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[w:Melatonin|Melatonin]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Megha, K.B., Arathi, A., Shikha, S. et al. Significance of Melatonin in the Regulation of Circadian Rhythms and Disease Management. Mol Neurobiol 61, 5541–5571 (2024). https://doi.org/10.1007/s12035-024-03915-0

Meliska, C. J., Martínez, L. F., López, A. M., Sorenson, D. L., Nowakowski, S., Kripke, D. F., Elliott, J., &amp; Parry, B. L. (2013). Antepartum depression severity is increased during seasonally longer nights: relationship to melatonin and cortisol timing and quantity. Chronobiology international, 30(9), 1160–1173. https://doi.org/10.3109/07420528.2013.808652

Pääkkönen,T., Leppäluoto, J., Ikäheimo, T., &amp; Rintamäki, H., Ruokonen, A., Hassi, J., &amp; Palinkas, L., (2008). Seasonal Levels of Melatonin, Thyroid Hormones, Mood, and Cognition Near the Arctic Circle. Aviation, space, and environmental medicine. 79. 695-9. https://doi.org/10.3357/ASEM.2148.2008. 

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
* Antepartum Depression Severity is Increased During Seasonally Longer Nights: Relationship to Melatonin and Cortisol Timing and Quantity
* Effect of controlled-release melatonin on sleep quality, mood, and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour
* Seasonal levels of melatonin, thyroid hormones, mood, and cognition near the Arctic Circle 
* Melatonin as a treatment for mood disorders: a systematic review
* Melatonin and agomelatine for preventing seasonal affective disorder
* Seasonal rhythms in mental health: a narrative scoping review of neurobiological mechanisms and therapeutic interventions   
* Melatonin Biosynthesis and regulation in reproduction 
* Seasonal changes in mood and behavior: A cluster analytic approach https://www.sciencedirect.com/science/article/abs/pii/0165178189900498 

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Mood]]</text>
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    <revision>
      <id>2830706</id>
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      <contributor>
        <username>Jtneill</username>
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      <text bytes="10519" sha1="1hrg4uhcnkgsbeyzras5iigkoku9oh5" xml:space="preserve">{{title|Melatonin and Seasonal Mood Changes:&lt;br&gt;What role does melatonin play in seasonal mood changes?}}
__TOC__
[[File:Melatonin.svg|center|thumb|350x350px|Figure 1. Melatonin Chemical Structure]]
==Overview==

{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Case Study: Seasonal Levels of Melatonin near the Arctic Circle

Through explorations of the Arctic circle and the effects on mood and cognitive performance has on explorers, a study was conducted on why this would be the case. 

Circumpolar environments have been known for its changes in day night cycles as well as its temperature. 

Explorers within these research stations have reported increases in depressive symptoms. 

There was a hypothesis on being a link between melatonin production and the effects of the Arctic's seasonal characteristics. Especially the link of the light intensity and its suppression in producing melatonin. 


{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.
*

{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* How is Melatonin produced?
* What seasonal effects have been shown on mood?
* Does irregular melatonin production have an impact on Seasonal Mood Disorders and if so why?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between weather and criminal behaviour? (closed-ended)
* What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Melatonin==
[[w:Melatonin|Melatonin]] is a hormone, best know through its function of regulating circadian rhythms. (Megha, et al, 2024)

* Indolamines, a family of neurotransmitters, that share a common molecular structure
* Produced in the Pineal Gland 
** For vertabres melatonin is produced in the dark
** Melatonin is a key signaling molecule that connects changes in external light conditions with changes
*** After the retina receives a light signal, it transmits the signal to the pineal gland which then transforms the light signal into melatonin.
*Circulars through the blood stream 
*Various tissues and oragans can secrete melatonin 
**but only the reinal tissue of the pineal glad and eyes show periodic secretory activity
**linking the biosysynthesis of melatonin shows that circidian rhythm and seasonal characteristics associated with external light conditions
** Meliska et al (2013)

=== Functions ===

* Sleep (regulates circadian changes)
**
*linked with light
*proctective effects on the nervous system
*Broad spectrum and antibacterial and immunoregularory functions 
*Even looking into melatonin's relations with fertility and reproduction 

=== Melatonin Synthesis and secretion ===
[[File:Melatonin mechanism.jpg|thumb|Figure 2. Mechanism of melatonin biosynthesis]]

* tryptohan excreted from the small instestine into the blood circulation and actively taken up by pineal cells
* AANAT converts the tryptohan hydroxlase into ASMT 
* catalyzes into melatonin
* cAMP-catalyzed proteins promotates formation of a complex 
* reduces the K (m) of serotonin, enhancing melatonin production
* Melatonin is released into the blood and spinal fluid
* blinds into plasma proteins
* Distructed throughout most tissues 

[[File:Melatonin production in 24 hour cycle.jpg|center|thumb|431x431px|Figure 3. Melatonin production in 24 hour cycle]]

=== Melatonin Fluctuations/Imbalance ===

* Melatonin release cycles over time 
* Temperature
* Circandian cycle and daylight changes

&lt;quiz display=&quot;simple&quot;&gt;
{Melatonin is produced by the?
|type=&quot;()&quot;}
+ Pineal Gland
- Cerebral Cortex
- Brain Stem 
- Retina 
 &lt;/quiz&gt;

== Seasonal Mood Disorders ==
Seasonal Affective Disorder (SAD)

* Part of the depression and bipolar family
*

What are they 

Why are they 

== Studies ==

* Artic Study on melatonin and light 
* Effect on controlled-release melatonin on sleep quality, mood and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour (2003)
** Double-blind trial
*** Group A Melatonin 
*** Group B placebo 
** Connected with the photoperiod not the weather-related aspect of the seasons 

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[w:Melatonin|Melatonin]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Megha, K.B., Arathi, A., Shikha, S. et al. Significance of Melatonin in the Regulation of Circadian Rhythms and Disease Management. Mol Neurobiol 61, 5541–5571 (2024). https://doi.org/10.1007/s12035-024-03915-0

Meliska, C. J., Martínez, L. F., López, A. M., Sorenson, D. L., Nowakowski, S., Kripke, D. F., Elliott, J., &amp; Parry, B. L. (2013). Antepartum depression severity is increased during seasonally longer nights: relationship to melatonin and cortisol timing and quantity. Chronobiology international, 30(9), 1160–1173. https://doi.org/10.3109/07420528.2013.808652

Pääkkönen,T., Leppäluoto, J., Ikäheimo, T., &amp; Rintamäki, H., Ruokonen, A., Hassi, J., &amp; Palinkas, L., (2008). Seasonal Levels of Melatonin, Thyroid Hormones, Mood, and Cognition Near the Arctic Circle. Aviation, space, and environmental medicine. 79. 695-9. https://doi.org/10.3357/ASEM.2148.2008. 

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
* Antepartum Depression Severity is Increased During Seasonally Longer Nights: Relationship to Melatonin and Cortisol Timing and Quantity
* Effect of controlled-release melatonin on sleep quality, mood, and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour
* Seasonal levels of melatonin, thyroid hormones, mood, and cognition near the Arctic Circle 
* Melatonin as a treatment for mood disorders: a systematic review
* Melatonin and agomelatine for preventing seasonal affective disorder
* Seasonal rhythms in mental health: a narrative scoping review of neurobiological mechanisms and therapeutic interventions   
* Melatonin Biosynthesis and regulation in reproduction 
* Seasonal changes in mood and behavior: A cluster analytic approach https://www.sciencedirect.com/science/article/abs/pii/0165178189900498 

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Mood]]
[[Category:Motivation and emotion/Book/Environment]]</text>
      <sha1>1hrg4uhcnkgsbeyzras5iigkoku9oh5</sha1>
    </revision>
    <revision>
      <id>2830712</id>
      <parentid>2830706</parentid>
      <timestamp>2026-09-03T07:47:36Z</timestamp>
      <contributor>
        <username>U3224236</username>
        <id>3106774</id>
      </contributor>
      <minor />
      <comment>Added to SAD section</comment>
      <origin>2830712</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11135" sha1="emqgccje9244va1qpwall50tuy0hmbh" xml:space="preserve">{{title|Melatonin and Seasonal Mood Changes:&lt;br&gt;What role does melatonin play in seasonal mood changes?}}
__TOC__
[[File:Melatonin.svg|center|thumb|350x350px|Figure 1. Melatonin Chemical Structure]]
==Overview==

{{RoundBoxTop|theme=3}}
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Use a captioned image to illustrate the scenario]]

; Case Study: Seasonal Levels of Melatonin near the Arctic Circle

Through explorations of the Arctic circle and the effects on mood and cognitive performance has on explorers, a study was conducted on why this would be the case. 

Circumpolar environments have been known for its changes in day night cycles as well as its temperature. 

Explorers within these research stations have reported increases in depressive symptoms. 

There was a hypothesis on being a link between melatonin production and the effects of the Arctic's seasonal characteristics. Especially the link of the light intensity and its suppression in producing melatonin. 


{{RoundBoxBottom}}

The Overview section should provide:
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box

Recommended length: 180 to 330 words.{{RoundBoxTop|theme=3}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* How is Melatonin produced?
* What seasonal effects have been shown on mood?
* Does irregular melatonin production have an impact on Seasonal Mood Disorders and if so why?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between weather and criminal behaviour? (closed-ended)
* What is the relationship between weather and criminal behaviour? (open-ended)
{{RoundBoxBottom}}

==Melatonin==
[[w:Melatonin|Melatonin]] is a hormone, best know through its function of regulating circadian rhythms. (Megha, et al, 2024)

* Indolamines, a family of neurotransmitters, that share a common molecular structure
* Produced in the Pineal Gland 
** For vertabres melatonin is produced in the dark
** Melatonin is a key signaling molecule that connects changes in external light conditions with changes
*** After the retina receives a light signal, it transmits the signal to the pineal gland which then transforms the light signal into melatonin.
*Circulars through the blood stream 
*Various tissues and oragans can secrete melatonin 
**but only the reinal tissue of the pineal glad and eyes show periodic secretory activity
**linking the biosysynthesis of melatonin shows that circidian rhythm and seasonal characteristics associated with external light conditions
** Meliska et al (2013)

=== Functions ===

* Sleep (regulates circadian changes)
*linked with light
*proctective effects on the nervous system
*Broad spectrum and antibacterial and immunoregularory functions 
*Even looking into melatonin's relations with fertility and reproduction 

=== Melatonin Synthesis and secretion ===
[[File:Melatonin mechanism.jpg|thumb|Figure 2. Mechanism of melatonin biosynthesis]]

* tryptohan excreted from the small instestine into the blood circulation and actively taken up by pineal cells
* AANAT converts the tryptohan hydroxlase into ASMT 
* catalyzes into melatonin
* cAMP-catalyzed proteins promotates formation of a complex 
* reduces the K (m) of serotonin, enhancing melatonin production
* Melatonin is released into the blood and spinal fluid
* blinds into plasma proteins
* Distructed throughout most tissues 

[[File:Melatonin production in 24 hour cycle.jpg|center|thumb|431x431px|Figure 3. Melatonin production in 24 hour cycle]]

=== Melatonin Fluctuations/Imbalance ===

* Melatonin release cycles over time 
* Temperature
* Circandian cycle and daylight changes

&lt;quiz display=&quot;simple&quot;&gt;
{1. Melatonin is produced by the?
|type=&quot;()&quot;}
+ Pineal Gland
- Cerebral Cortex
- Brain Stem 
- Retina 
 &lt;/quiz&gt;

== Seasonal Mood Disorders ==
Seasonal Affective Disorder (SAD)

* Part of the depression and bipolar family
*Combination of biological and mood disturbances through out different seasonal patterns
**Most commonly occurring during autumn and winter but cases have shown in the other seasons too ( Kurlanksik &amp; Ibay, 2012) 
*Light therapy has been shown to be effective for most patients showing clinical improvements within one to two weeks after the start of treatment ( Kurlanksik &amp; Ibay, 2012)  


== Studies ==

* Artic Study on melatonin and light 
* Effect on controlled-release melatonin on sleep quality, mood and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour (2003)
** Double-blind trial
*** Group A Melatonin 
*** Group B placebo 
** Connected with the photoperiod not the weather-related aspect of the seasons 

==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[w:Melatonin|Melatonin]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Kurlansik, Stuart &amp; Ibay, Annamarie. (2012). Seasonal affective disorder. American family physician. 86. 1037-41. 

Megha, K.B., Arathi, A., Shikha, S. et al. Significance of Melatonin in the Regulation of Circadian Rhythms and Disease Management. Mol Neurobiol 61, 5541–5571 (2024). https://doi.org/10.1007/s12035-024-03915-0

Meliska, C. J., Martínez, L. F., López, A. M., Sorenson, D. L., Nowakowski, S., Kripke, D. F., Elliott, J., &amp; Parry, B. L. (2013). Antepartum depression severity is increased during seasonally longer nights: relationship to melatonin and cortisol timing and quantity. Chronobiology international, 30(9), 1160–1173. https://doi.org/10.3109/07420528.2013.808652

Pääkkönen,T., Leppäluoto, J., Ikäheimo, T., &amp; Rintamäki, H., Ruokonen, A., Hassi, J., &amp; Palinkas, L., (2008). Seasonal Levels of Melatonin, Thyroid Hormones, Mood, and Cognition Near the Arctic Circle. Aviation, space, and environmental medicine. 79. 695-9. https://doi.org/10.3357/ASEM.2148.2008. 

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
* Antepartum Depression Severity is Increased During Seasonally Longer Nights: Relationship to Melatonin and Cortisol Timing and Quantity
* Effect of controlled-release melatonin on sleep quality, mood, and quality of life in subjects with seasonal or weather-associated changes in mood and behaviour
* Seasonal levels of melatonin, thyroid hormones, mood, and cognition near the Arctic Circle 
* Melatonin as a treatment for mood disorders: a systematic review
* Melatonin and agomelatine for preventing seasonal affective disorder
* Seasonal rhythms in mental health: a narrative scoping review of neurobiological mechanisms and therapeutic interventions   
* Melatonin Biosynthesis and regulation in reproduction 
* Seasonal changes in mood and behavior: A cluster analytic approach https://www.sciencedirect.com/science/article/abs/pii/0165178189900498 
* Ngiam, Alicia. (2022). Light Deprivation and Its Effects. https://www.researchgate.net/publication/365021664_Light_Deprivation_and_Its_Effects 

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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    <title>Motivation and emotion/Book/2026/Emotion dysregulation</title>
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__TOC__
==Overview==
{{RoundBoxTop|theme=2}}
[[File:Burnout At Work - Occupational Burnout.jpg|Burnout_At_Work_-_Occupational_Burnout|150px|right|thumb|'''Figure 1.''' Taylor feels frustrated after the submissions portal froze at a critical moment.]]

'''Scenario:  '''

Taylor is trying to upload an assignment for his statistics class last minute but the submissions portal freezes exactly as he presses “submit”. He watches the cursor’s loading icon spin again and again, likely because the portal is overwhelmed by other last minute submissions. 

Though Taylor still has 15 minutes before the deadline, he finds himself continuing to aggressively spam click the laptop trackpad and becoming consumed by a feeling of frustration. Taylor can feel his heart rate increase, jaw tighten and leg shake beneath the desk. 

Then the trackpad becomes unresponsive. Taylor shoves his laptop aside and drops his head onto the table, angrily questioning why nothing ever seems to go his way (see Figure 1). {{RoundBoxBottom}}

From moments of [[wikipedia:Happiness|happiness]] and excitement to [[wikipedia:Sadness|sadness]] or irritation, [[w:emotions|emotions]] are an integral aspect of being human, and it is expected that these emotions fluctuate as we navigate the complexity of our everyday lives. Emotions are neither ‘good’ nor ‘bad’, rather emotions have a specific [[wikipedia:Evolution|evolutionary]] purpose that has helped us survive, adapt and interact with others (Šimić et al., 2021). Nonetheless, there are times in which strong and overwhelming emotions can impair our daily functioning. This is called emotion dysregulation.

Emotion dysregulation can be simply defined as the difficulty to cope with intense emotions, to the extent that an individual is unable to implement adaptive [[wikipedia:Coping|coping strategies]] (Gross &amp; Thompson, 2007, as cited in Aslan et al., 2024). Consequently, individuals may struggle with abrupt changes in [[wikipedia:Mood_(psychology)|mood]], [[wikipedia:Impulsivity|impulsive behaviour]] and emotional responses out of proportion to the situation (''reference''). Research has indicated that frequent experiences of emotion dysregulation can negatively affect our [[wikipedia:Well-being|well-being]] and may be a risk factor for the development and maintenance of [[w:Mental_health_disorders|mental health disorders]] (Beauchaine &amp; Cicchetti, 2019; Aslan et al., 2024).

Taylor’s disproportionate [[wikipedia:Anger|anger]] response, in the scenario above, conveys a moment of emotion dysregulation. His escalating frustration led to impulsive actions that can be considered excessive relative to the situation.

This chapter will investigate how psychological science offers empirical frameworks to guide our understanding of emotion dysregulation, its consequences and strategies we can use to support [[w:Emotional_self-regulation|emotion regulation]]. Through understanding these concepts, we can be better prepared to identify and respond to emotion dysregulation in adaptive ways.

'''(''Author note:'' will reword the last paragraph, and find missing reference in paragraph two).'''

{{RoundBoxTop|theme=2}}'''Focus questions'''
1.	How can emotion dysregulation be understood?

2.	How can psychological models help explain experiences of emotion dysregulation?

3.	How does emotion dysregulation impact daily functioning?

4.	What is the association between emotion dysregulation and psychological disorders?

5.	What approaches can help manage emotion dysregulation? {{RoundBoxBottom}}

==Understanding emotion dysregulation ==

* Emotions can be described as complex and dynamic psychological states that integrate subjective experiences, physiological responses and behavioural responses (Hockenbury &amp; Hockenbury, 2007 as cited in D’Agostino et al., 2017).
* Emotion regulation is the ability to understand and cope with the onset, intensity and expression of these emotional states (Grecucci et al., 2020).
** Aldao and colleagues (2010) conceptualised emotion regulation as the “processes through which individuals modulate their emotions consciously and subconsciously to respond to environmental demands”.
* Individuals usually develop more effective emotion regulation strategies with age (Kaufman et al., 2026). Nonetheless, impairments in emotion regulation are observed at all stages of development (Kaufman et al., 2026).

So, what happens when we cannot regulate our emotions?

* Despite the term being incorporated into vernacular, there is still much conceptual ambiguity surrounding the definition of emotion dysregulation due to its complexity. D’Agostino and colleagues (2017) identified five overlapping dimensions of emotion dysregulation that appear across research; reduced emotional awareness, decreased emotional reactivity, intense experiences and expression of emotion, emotional rigidity, and impaired cognitive reappraisals.
* Comparatively, Beauchaine (2015, p.876) posits that emotion dysregulation constitutes “a pattern of emotional experience and/or expression that interferes with appropriate goal-directed behavior”.
* These conceptualisations of emotion dysregulation emphasise deficits in emotion recognition, inability to inhibit impulsive emotional reactions, and the subsequent challenge to respond in ways that support goal attainment.  
* Alongside its impact on general well-being, pervasive emotion dysregulation is considered an important factor in the development and maintenance of psychological disorders (Yalvaç &amp; Gaynor, 2021; Beauchaine &amp; Dante Cicchetti, 2019).

(''Author note:'' will include another section on outlining the core features/signs of emotion dysregulation. May delve into the different maladaptive strategies characteristic of emotion dysregulation. Then will include a sentence on using psychological frameworks to help our understanding.)

=== Gross's extended process model (EPM) of emotion regulation ===

* Gross’s [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|extended process model of emotion regulation]] is grounded in appraisal theory, positing that emotions emerge from the cognitive evaluations individuals make in response to events (Grecucci et al., 2020). This model emphasises a process-orientated framework for emotion [dys]regulation (Gross, 2015).

* Gross (2015) defines emotion regulation as “a particular type of interaction between valuation systems”.

* Gross (2015) proposes three distinct stages of emotion regulation; identification, selection and implementation. Emotion dysregulation may occur at any of these stages.
* Sheppes et al. (2015):
** Identification-stage failures: difficulty reading emotional cues, leading to challenges initiating emotion regulation strategies and/or overidentification of emotions.
** Selection-stage failures: difficulties selecting an appropriate regulation strategy and/or a lack of access to adaptive strategies.
** Implementation-stage failures: impaired ability to carry-out a selected regulation strategy.
* Comment on the limitations of this model. 
** e.g., focus on emotion regulation rather than emotion dysregulation?

=== Linehan's biosocial model ===

* Linehan’s (1993) biosocial model proposes that emotion dysregulation constitutes the dynamic interactions between an individual’s biological emotional vulnerability and an invalidating environment during development. This model was initially developed within the context of [[wikipedia:Borderline_personality_disorder|Borderline Personality Disorder]].[[File:DBT Biosocial model.png|thumb|304x304px|'''Figure 2.''' Linehan's (1993) biosocial model of emotion dysregulation. ]]
* Emotional vulnerability refers to the genetic predisposition toward emotional hypersensitivity, hyperreactivity and long-lasting emotional reactions (Bemmouna &amp; Weiner, 2023). The [[wikipedia:Prefrontal_cortex|prefrontal cortex]] and [[wikipedia:Amygdala|amygdala]] are brain regions often implicated in the genetic disruption of emotional processing, contributing to increased emotional vulnerability (Bemmouna &amp; Weiner, 2023). 
* Invalidating environments are characterised as insufficient environmental responses to a child’s emotional needs, thereby the child does not learn to understand, recognise or appropriately react to emotional responses (Cronwell et al., 2009). 
* Linehan (1993) argues that an individual is more likely to develop pervasive emotion dysregulation if they have an emotionally sensitive temperament and receive persistent invalidating responses. Individuals may feel compelled to escalate their emotional reactions to communicate their unmet needs (Linehan, 1993).
* Linehan (1993) suggests that emotion dysregulation will lead to maladaptive response patterns when individuals are faced with challenging experiences.
* Comment on the limitations of this model (e.g., developed based on BPD?)

=== Gratz and Roemer's multidimensional model of emotion regulation ===

* Gratz and Roemer’s (2004) multidimensional model of emotion regulation aimed to provide a comprehensive and integrative approach to the conceptualisation and measurement of emotion regulation. 
* Using on this model, Gratz and Roemer (2004) developed the Difficulties in Emotion Regulation Scale (DERS) to assess emotion dysregulation. Based on the results, Gratz and Roemer (2004) conceptualised six core dimensions that characterise emotion dysregulation; 
*# lack of awareness of emotions 
*# lack of clarity of emotions
*# nonacceptance of emotional responses
*# inability to access regulation strategies
*# difficulties controlling impulsive behaviour when experiencing negative emotions
*# inability to engage in goal directed behaviour when experiencing negative emotions
* DERS is an empirically validated psychological assessment tool used to assess emotion dysregulation among adolescents and adults (Kaufman et al., 2026). In 2026, Kaufman and colleagues developed a short form version called DERS-SF.

{{robelbox|theme=3|title=Let's do a quick knowledge check!|icon=Paomedia small-n-flat light-bulb.svg|iconwidth=68px}}&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple header=none&gt;
{Which model links emotion dysregulation to emotional vulnerability and invalidating environments?
| type=&quot;(+)&quot;}
- Gross’s extended process model
+ Linehan’s biosocial model
- Gratz and Roemer’s multidimensional model

{Which model defines emotion dysregulation as “a particular type of interaction between valuation systems&quot;? 
| type=&quot;(+)&quot;}
- Linehan’s biosocial model
- Gratz and Roemer’s multidimensional model
+ Gross’s extended process model

{Which model aimed to develop an approach to conceptualise and measure emotion dysregulation?
| type=&quot;(+)&quot;}
+ Gratz and Roemer’s multidimensional model
- Gross’s extended process model
- Linehan’s biosocial model
}
&lt;/quiz&gt;
  &lt;/div&gt;
  {{Robelbox/close}}

==Impact of emotion dysregulation on daily functioning==

*Describe/argue how emotion dysregulation impacts individuals daily functioning – making sure to define daily functioning and why this could be problematic to overall well-being.
**Tani and colleagues (2015) identified that increased levels of emotion dysregulation were associated with lower satisfaction in couples’ relationship quality.
**Those that struggle to modulate their emotional responses often experience prolonged and more severe periods of distress (Boemo et al., 2022). 
**Samea and colleagues (2025) performed a meta-analysis on the relationship between emotion dysregulation and sleep deprivation. 
*Emphasise that emotion dysregulation negatively impacts multiple domains of daily functioning (decision-making, stress-responses, quality of life, etc.) – therefore addressing the importance of addressing emotion dysregulation. 

(''Author note:'' should this section be expanded using subheadings? For example, interpersonal conflict, impaired academic/occupational performance, and impact on general health/well-being. Need to make sure enough literature is available with non-clinical populations - or maybe I can include research from clinical populations?)

==Association between psychological disorders and emotion dysregulation==

*Alongside its impact on everyday functioning, emotion dysregulation is increasingly recognised as     a transdiagnostic factor in the development and maintenance of psychological disorders (Yalvaç &amp; Gaynor, 2021; Beauchaine &amp; Cicchetti, 2019).
*Transdiagnostic factors refer to underlying risk, maintenance or protective factors that are implicated across a diverse range psychological disorders, transcending diagnostic categories (Dalgeish et al., 2020).
*Beauchaine and Cicchetti (2019) observed that emotion dysregulation has been associated with [[wikipedia:Internalizing_disorder|internalizing disorders]], [[wikipedia:Externalizing_disorder|externalizing disorders]], [[wikipedia:Personality_disorder|personality disorders]] and [[wikipedia:Psychosis|psychotic disorders]]. Further, impairments in top-down processing of emotional reactivity are evident across many psychological disorders (Beauchaine &amp; Cicchetti, 2019).
*Emotion dysregulation in children and adolescents may be a predisposing factor to the emergence of psychological disorders in adulthood (Cole et al. 2017).

=== Emotion dysregulation in borderline personality disorder (BPD) ===

* Borderline personality disorder (BPD) is a psychological disorder defined by an enduring pattern of emotion dysregulation, impaired interpersonal functioning and an unstable sense of self (Bohus et al., 2021).
* Refer back to Linehan’s (1993) biosocial model ---&gt; developed for BPD.
** According to Linehan (1933), emotion dysregulation is a core feature of borderline personality disorder, accounting for most of its symptomology.
* Borderline personality disorder is associated with low emotional awareness, persistent [[wikipedia:Negative_affectivity|negative affect]] and the use of ineffective strategies to regulate emotions (Fitzpatrick et al., 2023).
* Among those with borderline personality disorder, impulsivity and dysfunctional behaviour is associated with experiences of increased emotional distress (Bohus et al., 2021).   
(''Author note:'' will expand upon the role of emotion dysregulation as the driving factor of BPD symptomology (e.g., risk or maintaining factor) and touch on how individuals with BPD use strategies that elicit short-term relief but perpetuate difficulties over time.) 

=== Emotion dysregulation in bipolar disorder (BD) ===

* [[wikipedia:Bipolar_disorder|Bipolar disorder]] is a psychological disorder associated with extreme changes in mood, energy and activity levels (Singh et al., 2025). Bipolar disorder is defined by recurrent episodes of [[wikipedia:Mania|mania]] or [[wikipedia:Hypomania|hypomania]] and [[wikipedia:Depression_(mood)|depression]] (Singh et al., 2025).
* Difficulties in emotion regulation are correlated with depressive and (hypo)manic episodes (Oliva et al., 2023).
* Those with bipolar disorder exhibit a reduced capacity to recognise and accept their emotions during both acute (hypo)manic and depressive episodes (Oliva et al., 2023).
* M’Bailara and colleagues (2009) found that even during [[wikipedia:Euthymia_(medicine)|euthymia]], participants with bipolar disorder experienced greater emotional reactivity and emotional intensity than control participants. This suggests that emotion dysregulation persists beyond acute symptoms and may account for the increased vulnerability to minor stressful events observed among those with bipolar (M’Bailara et al., 2009).

(''Author note:'' will expand upon the role of emotion dysregulation in BD symptomology (e.g., risk or maintaining factor) and touch on how individuals with BD use maladaptive emotion regulation strategies. Additionally, will look for a more recent study on euthymia and emotion dysregulation). 

=== Emotion dysregulation in attention-deficit hyperactivity disorder (ADHD) ===

* [[wikipedia:Attention_deficit_hyperactivity_disorder|Attention-deficit hyperactivity disorder]] (ADHD) is [[wikipedia:Neurodevelopmental_disorder|neurodevelopmental disorder]] defined by persistent patterns of inattention, hyperactivity and impulsivity (Bodalski et al., 2019). 
* Emerging research has indicated that emotion regulation difficulties in ADHD cannot be explained fully by the presence of [[wikipedia:Comorbidity|comorbid]] disorders, rather emotion dysregulation itself may be a distinct feature of ADHD (Bodalski et al., 2019). 
* In 2014, Bunford and colleagues found that emotion dysregulation predicted social impairments among adolescents with ADHD, particularly emotional excitability, impulsivity and prolonged emotional responses.
* Barkley (1997) proposed the executive functioning theory of ADHD, arguing that ADHD can be attributed to impairments in the behavioural inhibition processes that govern self-regulation and goal-directed behaviour.
** Disruptions in emotional inhibition processes are among the executive functions implicated in ADHD, contributing to emotion dysregulation (Mitchell et al., 2012).
(''Author note:'' will expand upon the role of emotion dysregulation in ADHD and touch on how individuals with ADHD use maladaptive emotion regulation strategies. Additionally, will give examples on how emotion dysregulation presents in ADHD). 

{{RoundBoxTop|theme=2}}'''Scenario: Emotional dysregulation in ADHD'''
{{em|(Author note: will incorporate a scenario (and figure) to explain the role of emotion dysregulation in ADHD.)}}{{RoundBoxBottom}}

== Approaches to managing emotion dysregulation ==

*Research indicates that emotion regulation is a learned skill that develops with practice over the lifespan (Wright et al., 2025).
**[[File:Practicing mindfulness promotes creativity.png|thumb|180x180px|'''Figure 3.''' Emotion regulation skills promote general well-being.   ]]Individuals emotion regulation capabilities are shaped through learning processes, including parental modelling and [[wikipedia:Co-regulation|co-regulation]] during childhood (Wright et al., 2025)

* [[wikipedia:Psychotherapy|Psychotherapy]] interventions or management strategies that build emotion regulation skills are integral to reducing emotion dysregulation and maladaptive strategies among those with psychological disorders.
* Outside of professional support, there are a range of practical everyday strategies that can assist in reducing emotion dysregulation and promoting general well-being (see Figure 3).  

=== Dialectical behaviour therapy (DBT) approaches to emotion dysregulation ===

* [[Motivation and emotion/Book/2025/Dialectical behaviour therapy and emotion regulation|Dialectical behaviour therapy (DBT)]] emerged as a psychological treatment approach rooted in behaviourism to address self-harm behaviours in borderline personality disorder (Linehan &amp; Wilks, 2015). 
* DBT focuses on developing practical skills across the four modules of [[wikipedia:Mindfulness|mindfulness]], interpersonal effectiveness, emotional regulation and distress tolerance (Linehan &amp; Wilks, 2015). 
* Abundant empirical research found that DBT was clinically relevant in reducing emotion dysregulation and maladaptive coping strategies among those with borderline personality disorder (Lenz et al., 2016). 
** Emotional regulation is considered a key mechanism of change in DBT outcomes (Lenz et al., 2016).
* DBT has broader clinical applications – will incorporate research on DBT’s effectiveness across other psychological disorders in which emotion dysregulation is a key component. 
* Comment on how DBT relates to/addresses to both Linehan’s biosocial model and Gross’s extended process model.

(''Author note'': will further expand upon the emotion dysregulation skills-training involved in DBT and further emphasis will be placed on evidence from literature.) 

=== Cognitive behavioural therapy (CBT) approaches to emotion dysregulation ===

* [[File:Beck's CognitiveTriad.png|thumb|279x279px|'''Figure 4.''' Representation of Beck's (1976) cognitive triad. ]][[wikipedia:Cognitive_behavioral_therapy|Cognitive behavioural therapy (CBT)]] is grounded on the premise that thoughts, behaviours and emotions are intrinsically intertwined (Preece &amp; Gross, 2026).

*Cognitive reappraisal is an important technique used in CBT to regulate emotions (Wang &amp; Yin, 2023).
*Preece and Gross (2026) posit that CBT should be understood through the lens of emotion regulation.
*Compare/link back to Gross's (2015) extended process model of emotion regulation and reference Beck’s (1976) cognitive triad (see Figure 4).

=== Physiological approaches to managing emotion dysregulation  ===

* Briefly cover emotion regulation through exercise – include research on how exercise can reduce emotion dysregulation through physiological mechanisms.
** Thayer and Lane (2000) posited the neurovisceral integration model to account for the observed relationship between physiological feedback circuits and affective systems.
* Briefly cover [[Motivation and emotion/Book/2025/Guided meditation and emotion regulation|emotion regulation through meditation]]/mindfulness - technique used to elicit the relaxation response which facilitates emotion regulation through physiological mechanisms.
** Benson and colleagues (1974) relaxation response theory. 
** Emphasise the accessibility of meditation/mindfulness.  
==Conclusion==
* Summarise emotion dysregulation, including its core features, conceptualisation as multidimensional construct and how psychological models can guide our understanding (despite diverse approaches/definitions). 
* Emphasise the influence and impact of emotion dysregulation on our day-to-day lives. Comment on the effect of emotion dysregulation on our well-being. 
* Restate how increasing evidence highlights emotion dysregulation as a transdiagnostic factor across many psychological disorders, thereby emotion dysregulation has important clinical applications in the treatment of these disorders.
* Provide a practical summary of the strategies that can reduce dysregulation. Aim to encourage the reader to try implement these strategies themselves to improve emotion self-regulation skills. 
** Comment again on the importance of emotion regulation skills, particularly when confronted with challenging or distressing situations (leads to overall healthier behaviours and well-being).

==See also==

* [[Motivation and emotion/Book/2024/ADHD and emotional regulation|ADHD and emotional regulation]] (Book chapter, 2024)
* [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|Cognitive strategies and emotion regulation]] (Book chapter, 2025)
* [[Motivation and emotion/Book/2025/Dialectical behaviour therapy and emotion regulation|Dialectical behaviour therapy and emotion regulation]] (Book chapter, 2025)
* [[w:Emotional_self-regulation|Emotional self-regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|Extended process model of emotion regulation]] (Book chapter, 2026)

* [[Motivation and emotion/Book/2025/Social media and emotional dysregulation|Social media and emotional dysregulation]] (Book chapter, 2025)

==References==
{{Hanging indent|1=

Aldao, A., Nolen-Hoeksema, S., &amp; Schweizer, S. (2010). Emotion-regulation strategies across psychopathology: A meta-analytic review. {{em|Clinical psychology review, 30}}(2), 217–237. https://doi.org/10.1016/j.cpr.2009.11.004 

Aslan, I. H., Dorey, L., Grant, J. E., &amp; Chamberlain, S. R. (2024). Emotion regulation across psychiatric disorders. {{em|CNS spectrums, 29}}(3), 215–220. https://doi.org/10.1017/S1092852924000270

Barkley, R. A. (1997). Behavioral inhibition, sustained attention, and executive functions: Constructing a unifying theory of ADHD. {{em|Psychological Bulletin, 121}}(1), 65–94. https://doi.org/10.1037/0033-2909.121.1.65

Beauchaine, T. P. (2015). Future directions in emotion dysregulation and youth psychopathology. {{em|Journal of Clinical Child &amp; Adolescent Psychology, 44}}(5), 875–896. https://doi.org/10.1080/15374416.2015.1038827

Beauchaine, T. P., &amp; Cicchetti, D. (2019). Emotion dysregulation and emerging psychopathology: A transdiagnostic, transdisciplinary perspective. {{em|Development and Psychopathology, 31}}, 799–804. https://doi.org/10.1017/S0954579419000671 

Beck, A. T. (1976). {{em|Cognitive therapy and the emotional disorders}}. International Universities Press.

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Boemo, T., Nieto, I., Vazquez, C., &amp; Sanchez-Lopez, A. (2022). Relations between emotion regulation strategies and affect in daily life: A systematic review and meta-analysis of studies using ecological momentary assessments. {{em|Neuroscience and Biobehavioral Reviews, 139}}, Article 104747. https://doi.org/10.1016/j.neubiorev.2022.104747

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Kaufman, E. A., Xia, M., Fosco, G., Yaptangco, M., Skidmore, C. R., &amp; Crowell, S. E. (2016). The difficulties in emotion regulation scale short form (DERS-SF): Validation and replication in adolescent and adult samples. {{em|Journal of psychopathology and behavioral assessment, 38}}(3), 443–455. https://doi.org/10.1007/s10862-015-9529-3

Lenz, A. S., Del Conte, G., Hollenbaugh, K. M., &amp; Callendar, K. (2016). Emotional regulation and interpersonal effectiveness as mechanisms of change for treatment outcomes within a DBT program for adolescents. {{em|Counseling Outcome Research and Evaluation, 7}}(2), 73–85. https://doi.org/10.1177/2150137816642439 

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Linehan, M. M., &amp; Wilks, C. R. (2015). The course and evolution of dialectical behavior therapy. {{em|The American Journal of Psychotherapy, 69}}(2), 97-110. https://doi.org/10.1176/appi.psychotherapy.2015.69.2.97 

M’Bailara, K., Demotes-Mainard, J., Swendsen, J., Mathieu, F., Leboyer, M., &amp; Henry, C. (2009). Emotional hyper-reactivity in normothymic bipolar patients. {{em|Bipolar Disorders, 11}}(1), 63-69. https://doi.org/10.1111/j.1399-5618.2008.00656.x

Mitchell, J. T., Robertson, C. D., Anastopolous, A. D., Nelson-Gray, R. O., &amp; Kollins, S. H. (2012). Emotion dysregulation and emotional impulsivity among adults with attention-deficit/hyperactivity disorder: Results of a preliminary study. {{em|Journal of Psychopathology and Behavioral Assessment, 34}}(4), 510–519. https://doi.org/10.1007/s10862-012-9297-2

Oliva, V., De Prisco, M., Fico, G., Possidente, C., Fortea, L., Montejo, L., Anmella, G., Hidalgo-Mazzei, D., Grande, I., Murru, A., Fornaro, M., de Bartolomeis, A., Dodd, A., Fanelli, G., Fabbri, C., Serretti, A., Vieta, E., &amp; Radua, J. (2023). Correlation between emotion dysregulation and mood symptoms of bipolar disorder: A systematic review and meta-analysis. {{em|Acta psychiatrica Scandinavica, 148}}(6), 472–490. https://doi.org/10.1111/acps.1361 

Preece, D. A., &amp; Gross, J. J. (2026). Cognitive behavior therapy: An emotion regulation perspective. {{em|Clinical Psychology: Science and Practice}}. Advance online publication. https://doi.org/10.1037/cps0000327

Samea, F., Mortazavi, N., Reimann, G. M., Ebneabbasi, A., Zarei, M., Khazaie, H., Goldstein-Piekarski, A. N., Spiegelhalder, K., Baglioni, C., Sepehry, A. A., &amp; Tahmasian, M. (2025). Insomnia and emotion dysregulation: A meta-analytical perspective integrating regulatory strategies and dispositional difficulties. {{em|Sleep medicine reviews, 82}}, Article 102111. https://doi.org/10.1016/j.smrv.2025.102111 

Sheppes, G., Suri, G., &amp; Gross, J. J. (2015). Emotion regulation and psychopathology. {{em|Annual review of clinical psychology, 11}}, 379–405. https://doi.org/10.1146/annurev-clinpsy-032814-112739 

Šimić, G., Tkalčić, M., Vukić, V., Mulc, D., Španić, E., Šagud, M., Olucha-Bordonau, F. E., Vukšić, M., &amp; Hof, P. R. (2021). Understanding emotions: Origins and roles of the amygdala. {{em|Biomolecules, 11}}(6), Article 823. https://doi.org/10.3390/biom11060823

Singh, B., Swartz, H. A., Cuellar-Barboza, A. B., Schaffer, A., Kato, T., Dols, A., Sperry, S. H., Vassilev, A. B., Burdick, K. E., &amp; Frye, M. A. (2025). Bipolar disorder. {{em|The Lancet, 406}}(10506), 963–978. https://doi.org/10.1016/S0140-6736(25)01140-7 

Tani, F., Pascuzzi, D., &amp; Raffagnino, R. (2015). Emotion regulation and quality of close relationship: The effects of emotion dysregulation processes on couple intimacy. {{em|Applied Psychology Bulletin, 272}}(63), 3–15.

Thayer, J. F., &amp; Lane, R. D. (2000). A model of neurovisceral integration in emotion regulation and dysregulation. {{em|Journal of Affective Disorders, 61}}(3), 201-216. https://doi.org/10.1016/S0165-0327(00)00338-4. 

Wang, Y. X., &amp; Yin, B. (2023). A new understanding of the cognitive reappraisal technique: an extension based on the schema theory. {{em|Frontiers in behavioral neuroscience, 17}}, Article 1174585. https://doi.org/10.3389/fnbeh.2023.1174585 

Wright. R. N., Adock, R. A., &amp; LaBar, K. S. (2025). Learning emotion regulation: An integrative framework. {{em|The Psychological Review, 132}}(1), 172-203. https://doi.org/10.1037/rev0000506 

Yalvaç, E. B. K., &amp; Gaynor, K. (2021). Emotional dysregulation in adults: The influence of rumination and negative secondary appraisals of emotion. {{em|Journal of Affective Disorders, 282}}, 656-661. https://doi.org/10.1016/j.jad.2020.12.194

}}

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==External links==

* [https://www.healthline.com/health/how-to-control-your-emotions How To Become The Boss of Your Emotions] (Healthline)
* [https://www.ted.com/talks/ted_ed_how_to_manage_your_emotions How to manage your emotions] (TED-Ed)
* [https://www.youtube.com/watch?v=SWvHZkkrAjc How to Master Your Emotions &amp; Never Get Angry or Bothered by Anyone] (The Mel Robbins Podcast)
* [https://www.psychologytoday.com/au/blog/click-here-for-happiness/202108/what-is-emotional-dysregulation What Is Emotional Dysregulation?] (Psychology Today)

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__TOC__
==Overview==
{{RoundBoxTop|theme=2}}
[[File:Burnout At Work - Occupational Burnout.jpg|Burnout_At_Work_-_Occupational_Burnout|150px|right|thumb|'''Figure 1.''' Taylor feels frustrated after the submissions portal froze at a critical moment.]]

'''Scenario:  '''

Taylor is trying to upload an assignment for his statistics class last minute but the submissions portal freezes exactly as he presses “submit”. He watches the cursor’s loading icon spin again and again, likely because the portal is overwhelmed by other last minute submissions. 

Though Taylor still has 15 minutes before the deadline, he finds himself continuing to aggressively spam click the laptop trackpad and becoming consumed by a feeling of frustration. Taylor can feel his heart rate increase, jaw tighten and leg shake beneath the desk. 

Then the trackpad becomes unresponsive. Taylor shoves his laptop aside and drops his head onto the table, angrily questioning why nothing ever seems to go his way (see Figure 1). {{RoundBoxBottom}}

From moments of [[wikipedia:Happiness|happiness]] and excitement to [[wikipedia:Sadness|sadness]] or irritation, [[w:emotions|emotions]] are an integral aspect of being human, and it is expected that these emotions fluctuate as we navigate the complexity of our everyday lives. Emotions are neither ‘good’ nor ‘bad’, rather emotions have a specific [[wikipedia:Evolution|evolutionary]] purpose that has helped us survive, adapt and interact with others (Šimić et al., 2021). Nonetheless, there are times in which strong and overwhelming emotions can impair our daily functioning. This is called [[wikipedia:Emotional_dysregulation|emotion dysregulation]].

Emotion dysregulation can be simply defined as the difficulty to cope with intense emotions, to the extent that an individual is unable to implement adaptive [[wikipedia:Coping|coping strategies]] (Gross &amp; Thompson, 2007, as cited in Aslan et al., 2024). Consequently, individuals may struggle with abrupt changes in [[wikipedia:Mood_(psychology)|mood]], [[wikipedia:Impulsivity|impulsive behaviour]] and emotional responses out of proportion to the situation (''reference''). Research has indicated that frequent experiences of emotion dysregulation can negatively affect our [[wikipedia:Well-being|well-being]] and may be a risk factor for the development and maintenance of [[w:Mental_health_disorders|mental health disorders]] (Beauchaine &amp; Cicchetti, 2019; Aslan et al., 2024).

Taylor’s disproportionate [[wikipedia:Anger|anger]] response, in the scenario above, conveys a moment of emotion dysregulation. His escalating [[wikipedia:Frustration|frustration]] led to impulsive actions that can be considered excessive relative to the situation.

This chapter will investigate how psychological science offers [[wikipedia:Empirical_research|empirical frameworks]] to guide our understanding of emotion dysregulation, its consequences and strategies we can use to support [[w:Emotional_self-regulation|emotion regulation]]. Through understanding these concepts, we can be better prepared to identify and respond to emotion dysregulation in adaptive ways.

'''(''Author note:'' will reword the last paragraph, and find missing reference in paragraph two).'''

{{RoundBoxTop|theme=2}}'''Focus questions'''
1.	How can emotion dysregulation be understood?

2.	How can psychological models help explain experiences of emotion dysregulation?

3.	How does emotion dysregulation impact daily functioning?

4.	What is the association between emotion dysregulation and psychological disorders?

5.	What approaches can help manage emotion dysregulation? {{RoundBoxBottom}}

==Understanding emotion dysregulation ==

* Emotions can be described as complex and dynamic psychological states that integrate subjective experiences, physiological responses and behavioural responses (Hockenbury &amp; Hockenbury, 2007 as cited in D’Agostino et al., 2017).
* Emotion regulation is the ability to understand and cope with the onset, intensity and expression of these emotional states (Grecucci et al., 2020).
** Aldao and colleagues (2010) conceptualised emotion regulation as the “processes through which individuals modulate their emotions consciously and subconsciously to respond to environmental demands”.
* Individuals usually develop more effective emotion regulation strategies with age (Kaufman et al., 2026). Nonetheless, impairments in emotion regulation are observed at all stages of development (Kaufman et al., 2026).

So, what happens when we cannot regulate our emotions?

* Despite the term being incorporated into vernacular, there is still much conceptual ambiguity surrounding the definition of emotion dysregulation due to its complexity. D’Agostino and colleagues (2017) identified five overlapping dimensions of emotion dysregulation that appear across research; reduced emotional awareness, decreased emotional reactivity, intense experiences and expression of emotion, emotional rigidity, and impaired cognitive reappraisals.
* Comparatively, Beauchaine (2015, p.876) posits that emotion dysregulation constitutes “a pattern of emotional experience and/or expression that interferes with appropriate goal-directed behavior”.
* These conceptualisations of emotion dysregulation emphasise deficits in emotion recognition, inability to inhibit impulsive emotional reactions, and the subsequent challenge to respond in ways that support goal attainment.  
* Alongside its impact on general well-being, pervasive emotion dysregulation is considered an important factor in the development and maintenance of psychological disorders (Yalvaç &amp; Gaynor, 2021; Beauchaine &amp; Dante Cicchetti, 2019).

(''Author note:'' will include another section on outlining the core features/signs of emotion dysregulation. May delve into the different maladaptive strategies characteristic of emotion dysregulation. Then will include a sentence on using psychological frameworks to help our understanding.)

=== Gross's extended process model (EPM) of emotion regulation ===

* Gross’s [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|extended process model of emotion regulation]] is grounded in appraisal theory, positing that emotions emerge from the cognitive evaluations individuals make in response to events (Grecucci et al., 2020). This model emphasises a process-orientated framework for emotion [dys]regulation (Gross, 2015).

* Gross (2015) defines emotion regulation as “a particular type of interaction between valuation systems”.

* Gross (2015) proposes three distinct stages of emotion regulation; identification, selection and implementation. Emotion dysregulation may occur at any of these stages.
* Sheppes et al. (2015):
** Identification-stage failures: difficulty reading emotional cues, leading to challenges initiating emotion regulation strategies and/or overidentification of emotions.
** Selection-stage failures: difficulties selecting an appropriate regulation strategy and/or a lack of access to adaptive strategies.
** Implementation-stage failures: impaired ability to carry-out a selected regulation strategy.
* Comment on the limitations of this model. 
** e.g., focus on emotion regulation rather than emotion dysregulation?

=== Linehan's biosocial model ===

* Linehan’s (1993) biosocial model proposes that emotion dysregulation constitutes the dynamic interactions between an individual’s biological emotional vulnerability and an invalidating environment during development. This model was initially developed within the context of [[wikipedia:Borderline_personality_disorder|Borderline Personality Disorder]].[[File:DBT Biosocial model.png|thumb|304x304px|'''Figure 2.''' Linehan's (1993) biosocial model of emotion dysregulation. ]]
* Emotional vulnerability refers to the genetic predisposition toward emotional hypersensitivity, hyperreactivity and long-lasting emotional reactions (Bemmouna &amp; Weiner, 2023). The [[wikipedia:Prefrontal_cortex|prefrontal cortex]] and [[wikipedia:Amygdala|amygdala]] are brain regions often implicated in the genetic disruption of emotional processing, contributing to increased emotional vulnerability (Bemmouna &amp; Weiner, 2023). 
* Invalidating environments are characterised as insufficient environmental responses to a child’s emotional needs, thereby the child does not learn to understand, recognise or appropriately react to emotional responses (Cronwell et al., 2009). 
* Linehan (1993) argues that an individual is more likely to develop pervasive emotion dysregulation if they have an emotionally sensitive temperament and receive persistent invalidating responses. Individuals may feel compelled to escalate their emotional reactions to communicate their unmet needs (Linehan, 1993).
* Linehan (1993) suggests that emotion dysregulation will lead to maladaptive response patterns when individuals are faced with challenging experiences.
* Comment on the limitations of this model (e.g., developed based on BPD?)

=== Gratz and Roemer's multidimensional model of emotion regulation ===

* Gratz and Roemer’s (2004) multidimensional model of emotion regulation aimed to provide a comprehensive and integrative approach to the conceptualisation and measurement of emotion regulation. 
* Using on this model, Gratz and Roemer (2004) developed the Difficulties in Emotion Regulation Scale (DERS) to assess emotion dysregulation. Based on the results, Gratz and Roemer (2004) conceptualised six core dimensions that characterise emotion dysregulation; 
*# lack of awareness of emotions 
*# lack of clarity of emotions
*# nonacceptance of emotional responses
*# inability to access regulation strategies
*# difficulties controlling impulsive behaviour when experiencing negative emotions
*# inability to engage in goal directed behaviour when experiencing negative emotions
* DERS is an empirically validated psychological assessment tool used to assess emotion dysregulation among adolescents and adults (Kaufman et al., 2026). In 2026, Kaufman and colleagues developed a short form version called DERS-SF.

{{robelbox|theme=3|title=Let's do a quick knowledge check!|icon=Paomedia small-n-flat light-bulb.svg|iconwidth=68px}}&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple header=none&gt;
{Which model links emotion dysregulation to emotional vulnerability and invalidating environments?
| type=&quot;(+)&quot;}
- Gross’s extended process model
+ Linehan’s biosocial model
- Gratz and Roemer’s multidimensional model

{Which model defines emotion dysregulation as “a particular type of interaction between valuation systems&quot;? 
| type=&quot;(+)&quot;}
- Linehan’s biosocial model
- Gratz and Roemer’s multidimensional model
+ Gross’s extended process model

{Which model aimed to develop an approach to conceptualise and measure emotion dysregulation?
| type=&quot;(+)&quot;}
+ Gratz and Roemer’s multidimensional model
- Gross’s extended process model
- Linehan’s biosocial model
}
&lt;/quiz&gt;
  &lt;/div&gt;
  {{Robelbox/close}}

==Impact of emotion dysregulation on daily functioning==

*Describe/argue how emotion dysregulation impacts individuals daily functioning – making sure to define daily functioning and why this could be problematic to overall well-being.
**Tani and colleagues (2015) identified that increased levels of emotion dysregulation were associated with lower satisfaction in couples’ relationship quality.
**Those that struggle to modulate their emotional responses often experience prolonged and more severe periods of distress (Boemo et al., 2022). 
**Samea and colleagues (2025) performed a meta-analysis on the relationship between emotion dysregulation and sleep deprivation. 
*Emphasise that emotion dysregulation negatively impacts multiple domains of daily functioning (decision-making, stress-responses, quality of life, etc.) – therefore addressing the importance of addressing emotion dysregulation. 

(''Author note:'' should this section be expanded using subheadings? For example, interpersonal conflict, impaired academic/occupational performance, and impact on general health/well-being. Need to make sure enough literature is available with non-clinical populations - or maybe I can include research from clinical populations?)

==Association between psychological disorders and emotion dysregulation==

*Alongside its impact on everyday functioning, emotion dysregulation is increasingly recognised as     a transdiagnostic factor in the development and maintenance of psychological disorders (Yalvaç &amp; Gaynor, 2021; Beauchaine &amp; Cicchetti, 2019).
*Transdiagnostic factors refer to underlying risk, maintenance or protective factors that are implicated across a diverse range psychological disorders, transcending diagnostic categories (Dalgeish et al., 2020).
*Beauchaine and Cicchetti (2019) observed that emotion dysregulation has been associated with [[wikipedia:Internalizing_disorder|internalizing disorders]], [[wikipedia:Externalizing_disorder|externalizing disorders]], [[wikipedia:Personality_disorder|personality disorders]] and [[wikipedia:Psychosis|psychotic disorders]]. Further, impairments in top-down processing of emotional reactivity are evident across many psychological disorders (Beauchaine &amp; Cicchetti, 2019).
*Emotion dysregulation in children and adolescents may be a predisposing factor to the emergence of psychological disorders in adulthood (Cole et al. 2017).

=== Emotion dysregulation in borderline personality disorder (BPD) ===

* Borderline personality disorder (BPD) is a psychological disorder defined by an enduring pattern of emotion dysregulation, impaired interpersonal functioning and an unstable sense of self (Bohus et al., 2021).
* Refer back to Linehan’s (1993) biosocial model ---&gt; developed for BPD.
** According to Linehan (1933), emotion dysregulation is a core feature of borderline personality disorder, accounting for most of its symptomology.
* Borderline personality disorder is associated with low emotional awareness, persistent [[wikipedia:Negative_affectivity|negative affect]] and the use of ineffective strategies to regulate emotions (Fitzpatrick et al., 2023).
* Among those with borderline personality disorder, impulsivity and dysfunctional behaviour is associated with experiences of increased emotional distress (Bohus et al., 2021).   
(''Author note:'' will expand upon the role of emotion dysregulation as the driving factor of BPD symptomology (e.g., risk or maintaining factor) and touch on how individuals with BPD use strategies that elicit short-term relief but perpetuate difficulties over time.) 

=== Emotion dysregulation in bipolar disorder (BD) ===

* [[wikipedia:Bipolar_disorder|Bipolar disorder]] is a psychological disorder associated with extreme changes in mood, energy and activity levels (Singh et al., 2025). Bipolar disorder is defined by recurrent episodes of [[wikipedia:Mania|mania]] or [[wikipedia:Hypomania|hypomania]] and [[wikipedia:Depression_(mood)|depression]] (Singh et al., 2025).
* Difficulties in emotion regulation are correlated with depressive and (hypo)manic episodes (Oliva et al., 2023).
* Those with bipolar disorder exhibit a reduced capacity to recognise and accept their emotions during both acute (hypo)manic and depressive episodes (Oliva et al., 2023).
* M’Bailara and colleagues (2009) found that even during [[wikipedia:Euthymia_(medicine)|euthymia]], participants with bipolar disorder experienced greater emotional reactivity and emotional intensity than control participants. This suggests that emotion dysregulation persists beyond acute symptoms and may account for the increased vulnerability to minor stressful events observed among those with bipolar (M’Bailara et al., 2009).

(''Author note:'' will expand upon the role of emotion dysregulation in BD symptomology (e.g., risk or maintaining factor) and touch on how individuals with BD use maladaptive emotion regulation strategies. Additionally, will look for a more recent study on euthymia and emotion dysregulation). 

=== Emotion dysregulation in attention-deficit hyperactivity disorder (ADHD) ===

* [[wikipedia:Attention_deficit_hyperactivity_disorder|Attention-deficit hyperactivity disorder]] (ADHD) is [[wikipedia:Neurodevelopmental_disorder|neurodevelopmental disorder]] defined by persistent patterns of inattention, hyperactivity and impulsivity (Bodalski et al., 2019). 
* Emerging research has indicated that emotion regulation difficulties in ADHD cannot be explained fully by the presence of [[wikipedia:Comorbidity|comorbid]] disorders, rather emotion dysregulation itself may be a distinct feature of ADHD (Bodalski et al., 2019). 
* In 2014, Bunford and colleagues found that emotion dysregulation predicted social impairments among adolescents with ADHD, particularly emotional excitability, impulsivity and prolonged emotional responses.
* Barkley (1997) proposed the executive functioning theory of ADHD, arguing that ADHD can be attributed to impairments in the behavioural inhibition processes that govern self-regulation and goal-directed behaviour.
** Disruptions in emotional inhibition processes are among the executive functions implicated in ADHD, contributing to emotion dysregulation (Mitchell et al., 2012).
(''Author note:'' will expand upon the role of emotion dysregulation in ADHD and touch on how individuals with ADHD use maladaptive emotion regulation strategies. Additionally, will give examples on how emotion dysregulation presents in ADHD). 

{{RoundBoxTop|theme=2}}'''Scenario: Emotional dysregulation in ADHD'''
{{em|(Author note: will incorporate a scenario (and figure) to explain the role of emotion dysregulation in ADHD.)}}{{RoundBoxBottom}}

== Approaches to managing emotion dysregulation ==

*Research indicates that emotion regulation is a learned skill that develops with practice over the lifespan (Wright et al., 2025).
**[[File:Practicing mindfulness promotes creativity.png|thumb|180x180px|'''Figure 3.''' Emotion regulation skills promote general well-being.   ]]Individuals emotion regulation capabilities are shaped through learning processes, including parental modelling and [[wikipedia:Co-regulation|co-regulation]] during childhood (Wright et al., 2025)

* [[wikipedia:Psychotherapy|Psychotherapy]] interventions or management strategies that build emotion regulation skills are integral to reducing emotion dysregulation and maladaptive strategies among those with psychological disorders.
* Outside of professional support, there are a range of practical everyday strategies that can assist in reducing emotion dysregulation and promoting general well-being (see Figure 3).  

=== Dialectical behaviour therapy (DBT) approaches to emotion dysregulation ===

* [[Motivation and emotion/Book/2025/Dialectical behaviour therapy and emotion regulation|Dialectical behaviour therapy (DBT)]] emerged as a psychological treatment approach rooted in behaviourism to address self-harm behaviours in borderline personality disorder (Linehan &amp; Wilks, 2015). 
* DBT focuses on developing practical skills across the four modules of [[wikipedia:Mindfulness|mindfulness]], interpersonal effectiveness, emotional regulation and distress tolerance (Linehan &amp; Wilks, 2015). 
* Abundant empirical research found that DBT was clinically relevant in reducing emotion dysregulation and maladaptive coping strategies among those with borderline personality disorder (Lenz et al., 2016). 
** Emotional regulation is considered a key mechanism of change in DBT outcomes (Lenz et al., 2016).
* DBT has broader clinical applications – will incorporate research on DBT’s effectiveness across other psychological disorders in which emotion dysregulation is a key component. 
* Comment on how DBT relates to/addresses to both Linehan’s biosocial model and Gross’s extended process model.

(''Author note'': will further expand upon the emotion dysregulation skills-training involved in DBT and further emphasis will be placed on evidence from literature.) 

=== Cognitive behavioural therapy (CBT) approaches to emotion dysregulation ===

* [[File:Beck's CognitiveTriad.png|thumb|279x279px|'''Figure 4.''' Representation of Beck's (1976) cognitive triad. ]][[wikipedia:Cognitive_behavioral_therapy|Cognitive behavioural therapy (CBT)]] is grounded on the premise that thoughts, behaviours and emotions are intrinsically intertwined (Preece &amp; Gross, 2026).

*Cognitive reappraisal is an important technique used in CBT to regulate emotions (Wang &amp; Yin, 2023).
*Preece and Gross (2026) posit that CBT should be understood through the lens of emotion regulation.
*Compare/link back to Gross's (2015) extended process model of emotion regulation and reference Beck’s (1976) cognitive triad (see Figure 4).

=== Physiological approaches to managing emotion dysregulation  ===

* Briefly cover emotion regulation through exercise – include research on how exercise can reduce emotion dysregulation through physiological mechanisms.
** Thayer and Lane (2000) posited the neurovisceral integration model to account for the observed relationship between physiological feedback circuits and affective systems.
* Briefly cover [[Motivation and emotion/Book/2025/Guided meditation and emotion regulation|emotion regulation through meditation]]/mindfulness - technique used to elicit the relaxation response which facilitates emotion regulation through physiological mechanisms.
** Benson and colleagues (1974) relaxation response theory. 
** Emphasise the accessibility of meditation/mindfulness.  
==Conclusion==
* Summarise emotion dysregulation, including its core features, conceptualisation as multidimensional construct and how psychological models can guide our understanding (despite diverse approaches/definitions). 
* Emphasise the influence and impact of emotion dysregulation on our day-to-day lives. Comment on the effect of emotion dysregulation on our well-being. 
* Restate how increasing evidence highlights emotion dysregulation as a transdiagnostic factor across many psychological disorders, thereby emotion dysregulation has important clinical applications in the treatment of these disorders.
* Provide a practical summary of the strategies that can reduce dysregulation. Aim to encourage the reader to try implement these strategies themselves to improve emotion self-regulation skills. 
** Comment again on the importance of emotion regulation skills, particularly when confronted with challenging or distressing situations (leads to overall healthier behaviours and well-being).

==See also==

* [[Motivation and emotion/Book/2024/ADHD and emotional regulation|ADHD and emotional regulation]] (Book chapter, 2024)
* [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|Cognitive strategies and emotion regulation]] (Book chapter, 2025)
* [[Motivation and emotion/Book/2025/Dialectical behaviour therapy and emotion regulation|Dialectical behaviour therapy and emotion regulation]] (Book chapter, 2025)
* [[w:Emotional_self-regulation|Emotional self-regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|Extended process model of emotion regulation]] (Book chapter, 2026)

* [[Motivation and emotion/Book/2025/Social media and emotional dysregulation|Social media and emotional dysregulation]] (Book chapter, 2025)

==References==
{{Hanging indent|1=

Aldao, A., Nolen-Hoeksema, S., &amp; Schweizer, S. (2010). Emotion-regulation strategies across psychopathology: A meta-analytic review. {{em|Clinical psychology review, 30}}(2), 217–237. https://doi.org/10.1016/j.cpr.2009.11.004 

Aslan, I. H., Dorey, L., Grant, J. E., &amp; Chamberlain, S. R. (2024). Emotion regulation across psychiatric disorders. {{em|CNS spectrums, 29}}(3), 215–220. https://doi.org/10.1017/S1092852924000270

Barkley, R. A. (1997). Behavioral inhibition, sustained attention, and executive functions: Constructing a unifying theory of ADHD. {{em|Psychological Bulletin, 121}}(1), 65–94. https://doi.org/10.1037/0033-2909.121.1.65

Beauchaine, T. P. (2015). Future directions in emotion dysregulation and youth psychopathology. {{em|Journal of Clinical Child &amp; Adolescent Psychology, 44}}(5), 875–896. https://doi.org/10.1080/15374416.2015.1038827

Beauchaine, T. P., &amp; Cicchetti, D. (2019). Emotion dysregulation and emerging psychopathology: A transdiagnostic, transdisciplinary perspective. {{em|Development and Psychopathology, 31}}, 799–804. https://doi.org/10.1017/S0954579419000671 

Beck, A. T. (1976). {{em|Cognitive therapy and the emotional disorders}}. International Universities Press.

Bemmouna, D., &amp; Weiner, L. (2023). Linehan's biosocial model applied to emotion dysregulation in autism: A narrative review of the literature and an illustrative case conceptualization. {{em|Frontiers in psychiatry, 14}}, Article 1238116. https://doi.org/10.3389/fpsyt.2023.1238116 

Benson, H., Beary, J. F., &amp; Carol, M. P. (1974). The relaxation response. {{em|Psychiatry, 37}}(1), 37–46. https://doi.org/10.1080/00332747.1974.11023785 

Bodalski, E. A., Knouse, L. E., &amp; Kovalev, D. (2019). Adult ADHD, emotion dysregulation, and functional outcomes: Examining the role of emotion regulation strategies. {{em|Journal of Psychopathology Behavioral Assessment, 41}}, 81–92. https://doi.org/10.1007/s10862-018-9695-1 

Boemo, T., Nieto, I., Vazquez, C., &amp; Sanchez-Lopez, A. (2022). Relations between emotion regulation strategies and affect in daily life: A systematic review and meta-analysis of studies using ecological momentary assessments. {{em|Neuroscience and Biobehavioral Reviews, 139}}, Article 104747. https://doi.org/10.1016/j.neubiorev.2022.104747

Bohus, M., Stoffers-Winterling, J., Sharp, C., Krause, A. D., Schmahl, C., &amp; Lieb, K. (2021). Borderline personality disorder. {{em|The Lancet, 398}}(10310), 1528-1540. https://doi.org/10.1016/S0140-6736(21)00476-1 

Bunford, N., Evans, S. W., &amp; Langberg, J. M. (2014). Emotion dysregulation is associated with social impairment among young adolescents with ADHD. {{em|Journal of Attention Disorders, 22}}(1), 66–82. https://doi.org/10.1177/1087054714527793 

Cole, P. M., Hall, S. E., &amp; Hajal, N. J. (2017). Emotion dysregulation as a vulnerability to psychopathology. In T. P. Beauchaine and S. P. Hinshaw (Eds), {{em|Child and adolescent psychopathology}}, (3rd ed., 346-386). Wiley &amp; Sons. https://doi.org/10.1002/9781394258932.ch11

D’Agostino, A., Covanti, S., Rossi Monti, M., &amp; Starcevic, V. (2017). Reconsidering emotion dysregulation. {{em|Psychiatric Quarterly, 88}}, 807-825. https://doi.org/10.1007/s11126-017-9499-6 

Dalgleish, T., Black, M., Johnston, D., &amp; Bevan, A. (2020). Transdiagnostic approaches to mental health problems: Current status and future directions. {{em|Journal of consulting and clinical psychology, 88}}(3), 179–195. https://doi.org/10.1037/ccp0000482 

Fitzpatrick, S., Dixon-Gordon, K.L., Turner, C.J., Chen, S. X., &amp; Chapman, A. (2023). Emotion dysregulation in personality disorders. {{em|Current Psychiatry Reports, 24}}, 223–231. https://doi.org/10.1007/s11920-023-01418-8 

Gratz, K. L., &amp; Roemer, L. (2004). Multidimensional assessment of emotion regulation and dysregulation: Development, factor structure, and initial validation of the difficulties in emotion regulation scale. {{em|Journal of Psychopathology and Behavioral Assessment, 26}}, 41–54. https://doi.org/10.1023/B:JOBA.0000007455.08539.94 

Grecucci, A., Messina, I., Amodeo, L., Lapomarda, G., Crescentini, C., Dadomo, H., Panzeri, M., Theuninck, A., &amp; Frederickson, J. (2020). A dual route model for regulating emotions: Comparing models, techniques and biological mechanisms. {{em|Frontiers in Psychology, 11}}, Article 930. https://doi.org/10.3389/fpsyg.2020.00930

Gross, J. J. (2015) The extended process model of emotion regulation: Elaborations, applications, and future directions. {{em|Psychological Inquiry, 26}}(1), 130-137. https://doi.org/10.1080/1047840X.2015.989751 

Kaufman, E. A., Xia, M., Fosco, G., Yaptangco, M., Skidmore, C. R., &amp; Crowell, S. E. (2016). The difficulties in emotion regulation scale short form (DERS-SF): Validation and replication in adolescent and adult samples. {{em|Journal of psychopathology and behavioral assessment, 38}}(3), 443–455. https://doi.org/10.1007/s10862-015-9529-3

Lenz, A. S., Del Conte, G., Hollenbaugh, K. M., &amp; Callendar, K. (2016). Emotional regulation and interpersonal effectiveness as mechanisms of change for treatment outcomes within a DBT program for adolescents. {{em|Counseling Outcome Research and Evaluation, 7}}(2), 73–85. https://doi.org/10.1177/2150137816642439 

Linehan, M. M. (1993). {{em|Cognitive-behavioral treatment of borderline personality disorder}}. Guilford Press.

Linehan, M. M., &amp; Wilks, C. R. (2015). The course and evolution of dialectical behavior therapy. {{em|The American Journal of Psychotherapy, 69}}(2), 97-110. https://doi.org/10.1176/appi.psychotherapy.2015.69.2.97 

M’Bailara, K., Demotes-Mainard, J., Swendsen, J., Mathieu, F., Leboyer, M., &amp; Henry, C. (2009). Emotional hyper-reactivity in normothymic bipolar patients. {{em|Bipolar Disorders, 11}}(1), 63-69. https://doi.org/10.1111/j.1399-5618.2008.00656.x

Mitchell, J. T., Robertson, C. D., Anastopolous, A. D., Nelson-Gray, R. O., &amp; Kollins, S. H. (2012). Emotion dysregulation and emotional impulsivity among adults with attention-deficit/hyperactivity disorder: Results of a preliminary study. {{em|Journal of Psychopathology and Behavioral Assessment, 34}}(4), 510–519. https://doi.org/10.1007/s10862-012-9297-2

Oliva, V., De Prisco, M., Fico, G., Possidente, C., Fortea, L., Montejo, L., Anmella, G., Hidalgo-Mazzei, D., Grande, I., Murru, A., Fornaro, M., de Bartolomeis, A., Dodd, A., Fanelli, G., Fabbri, C., Serretti, A., Vieta, E., &amp; Radua, J. (2023). Correlation between emotion dysregulation and mood symptoms of bipolar disorder: A systematic review and meta-analysis. {{em|Acta psychiatrica Scandinavica, 148}}(6), 472–490. https://doi.org/10.1111/acps.1361 

Preece, D. A., &amp; Gross, J. J. (2026). Cognitive behavior therapy: An emotion regulation perspective. {{em|Clinical Psychology: Science and Practice}}. Advance online publication. https://doi.org/10.1037/cps0000327

Samea, F., Mortazavi, N., Reimann, G. M., Ebneabbasi, A., Zarei, M., Khazaie, H., Goldstein-Piekarski, A. N., Spiegelhalder, K., Baglioni, C., Sepehry, A. A., &amp; Tahmasian, M. (2025). Insomnia and emotion dysregulation: A meta-analytical perspective integrating regulatory strategies and dispositional difficulties. {{em|Sleep medicine reviews, 82}}, Article 102111. https://doi.org/10.1016/j.smrv.2025.102111 

Sheppes, G., Suri, G., &amp; Gross, J. J. (2015). Emotion regulation and psychopathology. {{em|Annual review of clinical psychology, 11}}, 379–405. https://doi.org/10.1146/annurev-clinpsy-032814-112739 

Šimić, G., Tkalčić, M., Vukić, V., Mulc, D., Španić, E., Šagud, M., Olucha-Bordonau, F. E., Vukšić, M., &amp; Hof, P. R. (2021). Understanding emotions: Origins and roles of the amygdala. {{em|Biomolecules, 11}}(6), Article 823. https://doi.org/10.3390/biom11060823

Singh, B., Swartz, H. A., Cuellar-Barboza, A. B., Schaffer, A., Kato, T., Dols, A., Sperry, S. H., Vassilev, A. B., Burdick, K. E., &amp; Frye, M. A. (2025). Bipolar disorder. {{em|The Lancet, 406}}(10506), 963–978. https://doi.org/10.1016/S0140-6736(25)01140-7 

Tani, F., Pascuzzi, D., &amp; Raffagnino, R. (2015). Emotion regulation and quality of close relationship: The effects of emotion dysregulation processes on couple intimacy. {{em|Applied Psychology Bulletin, 272}}(63), 3–15.

Thayer, J. F., &amp; Lane, R. D. (2000). A model of neurovisceral integration in emotion regulation and dysregulation. {{em|Journal of Affective Disorders, 61}}(3), 201-216. https://doi.org/10.1016/S0165-0327(00)00338-4. 

Wang, Y. X., &amp; Yin, B. (2023). A new understanding of the cognitive reappraisal technique: an extension based on the schema theory. {{em|Frontiers in behavioral neuroscience, 17}}, Article 1174585. https://doi.org/10.3389/fnbeh.2023.1174585 

Wright. R. N., Adock, R. A., &amp; LaBar, K. S. (2025). Learning emotion regulation: An integrative framework. {{em|The Psychological Review, 132}}(1), 172-203. https://doi.org/10.1037/rev0000506 

Yalvaç, E. B. K., &amp; Gaynor, K. (2021). Emotional dysregulation in adults: The influence of rumination and negative secondary appraisals of emotion. {{em|Journal of Affective Disorders, 282}}, 656-661. https://doi.org/10.1016/j.jad.2020.12.194

}}

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** Use &quot;Edit source&quot;
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==External links==

* [https://www.healthline.com/health/how-to-control-your-emotions How To Become The Boss of Your Emotions] (Healthline)
* [https://www.ted.com/talks/ted_ed_how_to_manage_your_emotions How to manage your emotions] (TED-Ed)
* [https://www.youtube.com/watch?v=SWvHZkkrAjc How to Master Your Emotions &amp; Never Get Angry or Bothered by Anyone] (The Mel Robbins Podcast)
* [https://www.psychologytoday.com/au/blog/click-here-for-happiness/202108/what-is-emotional-dysregulation What Is Emotional Dysregulation?] (Psychology Today)

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== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]]

'''Scenario'''

Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict.

Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively.

Why might people respond so differently to similar emotional situations?

One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing.

Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically.

Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship.

{{RoundBoxBottom}}

Emotional intelligence may play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences, maintain positive emotions, and support their overall emotional wellbeing. However, emotional intelligence is a complex construct, and different emotional abilities may contribute to wellbeing in different ways.
* '''How does emotional intelligence influence emotional wellbeing?'''  (Discuss the psychological processes that may explain the relationship. Emotional intelligence may help people recognise what they are feeling, understand why they are experiencing an emotion, regulate difficult emotions, cope with stress, and manage interpersonal situations. These processes may help explain why emotional intelligence is associated with wellbeing.)  '''Research needed:''' Research connecting emotional intelligence with emotion regulation, coping, stress management, and social functioning, as well as research linking these processes with wellbeing (Zomer, 2012; MacCann et al., 2022).
* '''Which components of emotional intelligence are particularly important for emotional wellbeing?'''  (Discuss whether different components of emotional intelligence contribute differently to wellbeing. Consider emotion perception, using emotion, emotion understanding, and emotion management/regulation. Emotion regulation may be particularly relevant, but this needs to be evaluated using research rather than assumed.)  '''Research needed:''' Studies comparing different dimensions or branches of emotional intelligence and their relationships with wellbeing, particularly emotion regulation/management (Blasco-Belled et al., 2019).
* '''How can emotional intelligence be developed to support emotional wellbeing?'''  (Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020).

* {{RoundBoxTop}}
; Focus questions
# What are emotional intelligence and emotional wellbeing? 
# What is the relationship between emotional intelligence and emotional wellbeing?
# How does emotional intelligence influence emotional wellbeing?
#Which components of emotional intelligence are particularly important for emotional wellbeing?
# How can emotional intelligence be developed to support emotional wellbeing?

{{RoundBoxBottom}}

== What are emotional intelligence and emotional wellbeing? ==
Before examining their relationship, it is important to establish what psychologists mean by emotional intelligence and emotional wellbeing. Both are multidimensional concepts, and different theoretical approaches influence how they are measured and understood.

=== Defining emotional intelligence ===
At its core, emotional intelligence concerns the processing and management of emotional information: noticing emotional cues, making sense of what they mean, and using that information to guide thought and behaviour (Givon et al., 2020). This is a somewhat different idea from how the term emotional is often used in everyday language. Describing someone as emotional usually refers to how frequently or intensely they display feelings, while emotional intelligence refers to how skilfully a person processes emotional information, regardless of how expressive they are (Luna et al., 2021). Similarly, EI should not be equated with simply being a nice person. Kindness reflects a disposition or value, whereas EI reflects an ability; a person can be highly emotionally intelligent while still setting firm boundaries or delivering difficult feedback, and a person can be warm and agreeable while struggling to accurately identify or regulate their own emotions (Salovey &amp; Grewal, 2005). Someone can therefore be outwardly calm and reserved while processing emotional information very effectively, just as someone can be outwardly expressive while struggling to understand or manage what they feel (Villanueva et al., 2020).

Because EI has been conceptualised in more than one way, it is useful to introduce these major approaches before going further. As already noted, ability models treat EI as a form of intelligence: a set of cognitive abilities for processing emotional information, assessed using performance-based tasks with objectively better or worse answers, in much the same way as other cognitive abilities are tested (Carfagno, 2020). Trait models instead focus on people's own perceptions of their emotional abilities and tendencies, typically measured through self-report questionnaires. Mixed models combine emotional abilities with broader personality-like characteristics, such as optimism, motivation, or assertiveness.

These differences in definition matter a great deal when examining research on EI and wellbeing. A study using a trait self-report measure is not necessarily capturing the same underlying construct as a study using a performance-based ability test; the two can produce meaningfully different results, including different-strength relationships with wellbeing outcomes (Barbash, 2015). Keeping this distinction in mind will be important later in this chapter, when comparing what the evidence shows across different EI models.

==== '''Ability model of emotional intelligence''' ====
{| class=&quot;wikitable&quot;
!Branch
!Description
|-
|'''Perceiving emotions'''
|Accurately identifying emotions in oneself and others, including through facial expressions, tone of voice, and body language.
|-
|'''Using emotions to facilitate thought'''
|Harnessing emotional information to guide attention, reasoning, and problem-solving.
|-
|'''Understanding emotions'''
|Comprehending emotional causes, consequences, and the way emotions change or combine over time.
|-
|'''Managing emotions'''
|Regulating one's own emotions and, where appropriate, the emotions of others, in ways that support goals and wellbeing.
|}
'''Table 1.''' ''The four-branch model''

Each branch can support wellbeing in its own way. Accurately noticing emotions helps people recognise what they and others feel. Using emotions to support thinking means letting feelings guide attention and judgment. Understanding emotions helps people figure out why emotions happen and how they change. Managing emotions is about handling emotional responses, which is especially helpful during tough times, like when Sophie takes a moment to calm herself before talking to her friend.

Explain the four branches:

# Perceiving emotions
# Using emotions to facilitate thought
# Understanding emotions[[File:Model of emotional intelligence.png|thumb|'''Figure 2.''' ''Mayer and Salovey's Emotional Intelligence model'' ]]Managing emotions

Explain briefly how each ability could theoretically contribute to wellbeing.

&lt;quiz display=&quot;simple&quot;&gt;
Quiz 1. 
Which statement best describes emotional intelligence?
|type=&quot;()&quot;}
+ The ability to perceive, understand, use, and manage emotions.
- The ability to avoid experiencing negative emotions.
- The ability to remain happy in all situations.
- The ability to control the emotions of other people.
}
&lt;/quiz&gt;

=== Trait emotional intelligence ===
* Explain trait EI.
* Distinguish trait EI from ability EI.
* Explain that trait EI concerns people's perceptions of their emotional abilities and tendencies.
* Briefly introduce why trait EI is relevant to emotional wellbeing.

=== Defining emotional wellbeing ===

** Define emotional wellbeing and explain what it means in psychology.
** Explain that emotional wellbeing involves both positive and negative emotional experiences.
** Clarify that good emotional wellbeing does not mean feeling happy or positive all the time.
** Discuss the ability to recognise, understand, and manage emotions as part of healthy emotional functioning.
** Explain that experiencing emotions such as sadness, anger, anxiety, or disappointment is a normal part of emotional wellbeing.
** Discuss how emotional wellbeing may involve being able to cope with difficult emotions and recover from stressful experiences (Chen et al., 2023).
** Distinguish emotional wellbeing from the absence of mental illness or psychological distress.
** Briefly explain how emotional wellbeing is measured in psychological research, as this will be important when you later discuss its relationship with emotional intelligence.

== What is the relationship between emotional intelligence and emotional wellbeing? ==
After defining both concepts, this section examines what psychological research shows about the overall relationship between emotional intelligence and emotional wellbeing.

=== Emotional intelligence and positive wellbeing ===

* Examine whether higher emotional intelligence is associated with greater emotional wellbeing.
* Discuss research involving positive affect, happiness, life satisfaction, and subjective or psychological wellbeing.
* Explain the strength of the relationship where research allows.
* Avoid assuming that emotional intelligence directly causes greater happiness.

=== Emotional intelligence and negative emotional experiences ===

* Examine the relationship between EI and stress.
* Discuss emotional distress and negative affect.
* Consider relevant research involving anxiety or depressive symptoms.
* Explain that higher EI does not mean that someone will never experience negative emotions.
* Instead, people with stronger emotional abilities may be better able to understand and respond to difficult emotions.

=== Trait and ability emotional intelligence ===

* Compare research findings for trait and ability EI.
* Examine whether trait EI shows different relationships with wellbeing than ability EI.
* Consider how the way EI is measured could influence research findings.
* Discuss the limitations of relying heavily on self-report measures.

=== What does the evidence tell us? ===
* Bring the research together.
* Establish whether there is consistent evidence for an association between EI and emotional wellbeing.
* Distinguish correlation from causation.
* Identify any inconsistencies or limitations in the evidence.

== How does emotional intelligence influence emotional wellbeing? ==
Finding an association between emotional intelligence and wellbeing does not necessarily explain why the two are related. Several psychological processes may help explain how emotional intelligence contributes to emotional wellbeing.

=== Emotional awareness and perception ===

* Explain the importance of accurately recognising emotions.
* Discuss how identifying an emotional state can help a person decide how to respond.
* Explain what may happen when people have difficulty recognising their emotions.
* Connect emotional awareness with emotional wellbeing.

=== Understanding emotions ===

* Discuss the ability to understand why an emotion has occurred.
* Explain how people can recognise changes and combinations of emotions.
* Consider how understanding the causes and consequences of emotions may support emotional functioning.
* Explain how emotional understanding may make it easier to choose an appropriate response.

=== Emotion regulation ===

* Define emotion regulation.
* Explain how EI may support the regulation of difficult emotional experiences.
* Discuss adaptive and maladaptive emotion regulation strategies.
* Consider cognitive reappraisal and suppression where relevant.
* Explain that regulating an emotion does not necessarily mean removing or suppressing it.
* Examine whether emotion regulation helps explain the relationship between EI and wellbeing.

=== Coping with stress ===

* Examine how EI may influence responses to stressful situations.
* Discuss appraisal and coping strategies.
* Consider emotional recovery and resilience.
* Connect effective coping with emotional wellbeing.

=== Social relationships and support ===

* Explain how EI may help people recognise other people's emotions.
* Discuss emotional communication.
* Examine interpersonal conflict and relationship management.
* Consider whether stronger relationships and social support provide another pathway between EI and wellbeing.

'''Figure 3. Possible pathways through which emotional intelligence may support emotional wellbeing.'''

Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, understanding, emotion regulation, coping with stress, and social functioning. These processes may work together rather than occurring in a simple step-by-step sequence.

== Which components of emotional intelligence are particularly important for emotional wellbeing? ==
Although emotional intelligence is often discussed as a single concept, its individual components may not contribute equally to emotional wellbeing.

=== Emotion perception ===

* Examine the benefits of accurately recognising emotional states.
* Discuss the importance of identifying emotional cues in oneself and others.
* Consider limitations: recognising an emotion does not necessarily mean someone will manage it effectively.

=== Using emotions ===

* Explain what it means to use emotions to support thinking.
* Discuss how emotional information may guide attention, judgement, or problem-solving.
* Examine whether this component has a clear relationship with emotional wellbeing.

=== Emotion understanding ===

* Discuss understanding the causes and consequences of emotions.
* Consider emotional complexity and changes in emotional states.
* Examine how understanding emotions may support coping and regulation.

=== Emotion regulation and management ===

* Examine whether emotion regulation is particularly strongly related to emotional wellbeing.
* Discuss positive affect and negative affect.
* Consider emotional distress.
* Examine connections with resilience and coping.
* Consider whether emotion regulation could help explain the broader relationship between EI and wellbeing.

=== Comparing the components ===

* Compare evidence across the different EI components.
* Consider whether any component appears particularly important for wellbeing.
* Avoid assuming that emotion regulation is the most important unless the research supports this conclusion.
* Consider whether the different abilities work together.

'''Table 2.'''

Emotional intelligence and emotional wellbeing

{| class=&quot;wikitable&quot;
!Concept
!Central concern
!Example
|-
|Emotional intelligence
|Processing, understanding and managing emotional information
|Recognising why you feel anxious and choosing an appropriate response
|-
|Emotional wellbeing
|Quality of emotional experience and functioning
|Experiencing manageable negative emotion while maintaining positive functioning
|-
|Emotion regulation
|Influencing emotional experiences or responses
|Reappraising a stressful situation rather than reacting impulsively
|}

== How can emotional intelligence be developed to support emotional wellbeing? ==
Discuss whether emotional intelligence can be improved through training or psychological interventions. Consider strategies aimed at emotional awareness, understanding emotions, emotion regulation, and interpersonal skills. Examine whether improving these abilities actually leads to improvements in wellbeing and acknowledge limitations in the evidence.)  '''Research needed:''' Intervention studies and preferably systematic reviews or meta-analyses of emotional intelligence training and its effects on emotional or psychological wellbeing (Hodzic et al., 2018); (Nadler et al., 2020)

== Figures ==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 3'''. Example of an image with a descriptive caption.]]
* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

;
;
'''Quizzes'''
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

==Conclusion==
* Emotional wellbeing involves healthy emotional experience and functioning rather than simply experiencing positive emotions.
* Research suggests that higher emotional intelligence is generally associated with better indicators of wellbeing.
* Emotion regulation, coping and interpersonal functioning may help explain this relationship.
* Different EI components and models may show different relationships with wellbeing.
* Emotional intelligence may be partly developable, creating potential applications for wellbeing interventions.
* or universal cause of emotional wellbeing.  Return briefly to Sophie:  Sophie's situation demonstrates that emotional intelligence does not remove disappointment, stress, or conflict. Instead, emotional abilities may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that support her wellbeing.  
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==

* [[Emotional intelligence]]
* Emotional regulation
* [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] {{ic|Use internal link style as shown in Tutorial 2}}
* Psychological wellbeing
* Coping
* Stress
* Positive psychology

* [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011)

* [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks)

* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
This section lists the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

'''APA style example:'''

{{Hanging indent|1=

Barbash, E. H. (2015). Emotional intelligence in professional psychology doctoral students: A cross-sectional study (Publication No. 3724176) [Doctoral dissertation, The Florida State University]. ProQuest Dissertations and Theses Global.https://www.proquest.com/psychology/docview/1725144779/3E4CE21A1D6E4E64PQ/1?accountid=28889&amp;sourcetype=Dissertations%20&amp;%20Theses

Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., &amp; Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x 

Carfagno, N. (2020). The factor structures of ability and trait emotional intelligences relative to general intelligence and personality. Proquest.Com. https://www.proquest.com/psychology/docview/2466049130/3E4CE21A1D6E4E64PQ/4?accountid=28889&amp;sourcetype=Dissertations%20&amp;%20Theses

Chen, C., Kotozaki , Y., Okubo , R., &amp; Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience
. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/

Givon, E., Itzhak-Raz, A., Danieli, G., Karmon-Presser, A., &amp; Meiran, N. (2020, March). How Does the Emotional Experience Evolve? Feeling Generation as Evidence Accumulation. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/3yqigdmzyv?route=details

Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., &amp; Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613

Llamas-Díaz, D., Cabello, R., Megías-Robles, A., &amp; Fernández-Berrocal, P. (2022). Systematic review and meta-analysis: The association between emotional intelligence and subjective well-being in adolescents. Journal of Adolescence, 94(7), 925–938. PubMed. https://doi.org/10.1002/jad.12075

Luna, L. M. B., Vilar, M. M., Soto, C. M., &amp; Santiago, J. L. C. (2021). Emotional intelligence measures: A systematic review. Healthcare, 9(12). https://doi.org/10.3390/healthcare9121696

Nadler, R., Carswell , J., &amp; Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/

Robinson, T. J., &amp; Zell, E. (2026). Robust associations of emotional intelligence with human flourishing: A second-order meta-analysis. Proceedings of the National Academy of Sciences of the United States of America, 123(19), e2532963123. https://doi.org/10.1073/pnas.2532963123

Salovey, P., &amp; Grewal, D. (2005). The Science of Emotional Intelligence. Current Directions in Psychological Science : A Journal of the American Psychological Society, 14(6), 281–285. https://doi.org/10.1111/j.0963-7214.2005.00381

MacCann, C., Double, K. S., &amp; Clarke, I. E. (2022). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Frontiers in Psychology, 13, 835819. https://doi.org/10.3389/fpsyg.2022.835819

Xu, X., Pang, W., &amp; Xia, M. (2021, December). Are emotionally intelligent people happier? A meta‐analysis of the relationship between emotional intelligence and subjective well‐being using chinese samples. Canberra.Edu.Au. https://research-ebsco-com.ezproxy.canberra.edu.au/c/aprr63/viewer/pdf/cwosfaeugr?route=details
Villanueva, L., Prado-Gascó, V., &amp; Montoya-Castilla, I. (2020). Longitudinal analysis of subjective well-being in preadolescents: The role of emotional intelligence, self-esteem and perceived stress. Journal of Health Psychology, 27(2), 135910532095160. https://doi.org/10.1177/1359105320951605

Zomer, L. (2012a). The relationships among emotional intelligence, gender, coping strategies, and well-being in the management of stress in close interpersonal relationships and the workplace [Master's thesis, University of Toronto]. https://www.proquest.com/openview/723018a77700f5c01e4992096ca8f1fa/1?pq-origsite=gscholar&amp;cbl=18750

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective]
* [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy]

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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__TOC__

== Overview ==
{{RoundBoxTop|theme=5}}
[[File:“Sex workers rights are human rights” - Europride 2019.jpg|thumb|200px|'''Figure 1'''. Prostitution in Sao Paulo, Brazil]]

'''What would you do if you found yourself in this scenario?'''

Feeling lost and left with little money of your own, but you have to make ends meet. What will you do when you have to pay your rent? What if you have a child? Would you turn to sex work to make ends meet? 

If you did... would that make your rights to a safe and trustworthy work environment just disappear? Sex work itself is illegal. Having been driven to such means to make ends meet, you must register as self-employed to pay taxes, buy a house, or even just be considered an employee. [https://doi.org/10.1080/14427591.2020.1830841.  (Huglstad et al., 2020)].

Sex workers rights activists proclaim that sex work is work, that they are not victims in need of help, and that they are entitled to any basic human right (see Figure 1). That in itself poses another scenario: What drives people into sex work? Is it finance and convenience? Or is it pleasure and fun?

{{RoundBoxBottom}}
* introduce sex work as a concept {{ic|Note to link to other work-related book chapters}}
* introduce the main driving forces that motivate people to go into sex work
* introduce influencing factors that influence worker experiences (whorephobia &amp; adult entertainment)
{{RoundBoxTop|theme=5}}
'''Focus questions'''
* What are the main driving forces causing people to turn to sex work?
* What is the effect of the adult entertainment industry on sex workers?
* What is the condition of work place satisfaction for sex workers?
* How does whorephobia and others' actions towards sex workers effect their work place?
* Do those who turn to sex work for pleasure have different experiences to those who do so for money alone?
{{RoundBoxBottom}}

== Driving forces pushing people towards [[Sex education|sex]] work ==
- Do those who turn to sex work for pleasure have different experiences to those who do so for money alone?

=== Age as a factor ===
- why do young people enter the sex work industry, how does that change workplace satisfaction?

- how does parenting influence age related sex work?

- do older individuals in the sex work industry experience more angism in their workplace?

=== Maslow's hierarchy of needs ===
- looking at maslows hierarchy of needs, what may cause people to turn to sex work?

- does peoples standing on the hierarchy influence their workplace satisfaction?

=== Socioeconomic factors ===
- is sex work more prevelent in low socioeconomic places/citys around the world?

- intrinsic vs extrinsic factors

- how does their treatment change from higher to lower socioeconomic places

== How motivations to join the sex work industry differ from men to women ==
{| class=&quot;wikitable&quot;
|+Percentages showing [[doi:10.1017/hyp.2024.56.|gender differences]] in paying for vs working in sex
!
!Participate in sex work
!Pay for sex work
|-
|Men
|20%
|12%
|-
|Women
|80%
|1%
|}

=== Male sex workers ===
- why are they less prevelent in the public eye?

- why are they not degraded as much as female workers?

- do men turn to sex work to fulfill their own physical needs?

=== Female sex workers ===
- why are they more prevelent in the public eye?

- why are they degraded more then male workers?

- do women turn to sex work to fulfill their own physical needs?

== The adult entertainment industry and its relationship to sex workers ==
- does this carer add unrealistic expectations towards sex workers and what people will ask to recieve?

- looking into detail about how the adult entertainment industry can glorify sex work and how that may motivate people to join the industry.

== Work place satisfaction for sex workers ==
- is there an element of sexual satisfaction which motivates sex workers?

- is there a difference between men and women?

- how to sexworkers respond to violence in their line of work and how may that motivate their actions?

== [https://www.jrc-concordia.ca/wp-content/uploads/2017/01/Journal-of-Religion-and-Culture-Vol-27-no-1.pdf#page=13 Whorephobia] and how it influences work place satisfaction ==
- how do unfavorable opinions of sex workers effect motivation and how does that relate to workplace satisfaction?

- how does the prevelence of this violence motivate sex workers to continue in their chosen carer path?

- how does whorephobia effect workplace satisfaction?&lt;quiz display=&quot;simple&quot;&gt;
{More men then women experience sexual violence as a sex worker:
|type=&quot;()&quot;}
- True
+ False

{There are more women in the sex trade then there are men:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
Through this chapter we have spoken about the different motivations that drives people into sex work and how that can influence work place experiences. We have done this by assesing how age and gender can influence motivations as well as what circumstances can effect those drives. 

- summerise the different motivations that lead to sex work, considering gender and age. 

- summerise how socioeconomic factors influence sex worker experiences, including prejudices. 
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[Maslow's hierarchy of needs]] (Wikiversity)
* [[Motivation and emotion/Book/2015/Sex worker motivation|Sex work motivation]] (Book chapter, 2015)

==References ==
{{Hanging indent|1=Arnold, K. A., Turner, N., Barling, J., &amp; Iverson, R. D. (2020). Feeling safe while doing sex work: Motivation for entering sex work moderates the relationship between perceptions of physical danger and desire to leave sex work. Safety science, 129, 104795.[https://pdf.sciencedirectassets.com/271730/1-s2.0-S0925753520X00063/1-s2.0-S0925753520301922/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEDgaCXVzLWVhc3QtMSJIMEYCIQDLwNUlkBLTVH2W9sUG%2FL8xBiniafqj2YjDThRYfM8j7gIhAPpb6usou%2FCfun2GlN94HXvsSkZFNyb27ndcS5LYkmOqKrIFCAEQBRoMMDU5MDAzNTQ2ODY1IgwVqu3mdLMxIbQQT4EqjwVyEzAMhHQLz8i1mRyce46T7kq8nFGMPfsWbvGbIjwD5sTztgejH2H6uoazHE3ceLn%2BHJAMU2gcjulIYqlqCSyvPuWe%2BScfjOHfBfQZnctFdUl2oWqaahIi9YzAVcTWeftoZk%2FylcCfLerGklhqD7HwVv2vZdBzsx94xmqQ1V1uUbIbAIkJ30NfSj00z1cSaUL1q96wM120kL%2By3Fd%2FYRnMyLzgn343yVJM%2BYWksKgOiezOVEa%2BhmIbNUYzcpb%2Fkcah8UWIgZWXjdLp2fZTJB2yCYbj091PqbNLxbKvPvqaCnGI%2BSTaCPpupfKIdR02m2jepxiZI8%2FZrLGy71f0vYzU%2Be120AQZZCPdjTZZAOP9xB%2Bvfpp48EyclSYmN0rSn2bZcMSiqosJl%2Bz6BiiW3J9SmbZXGNWeECGltEiG%2FP%2FRiZ0uB94c2nIGVzBsfJ4fe7UAD3mlWR9UBCT5FirlPE3Dzap%2BH1xnOe%2BGN481xmH1YSoL3FACDW8J5HUez%2BURv5vTzFDq4nON4cFWhMdgPNfu8v%2BjU35utqWccWAfsows5CFE%2FdiNDUQa7EI1XwF00wtRp2k06DjOWBl%2FggL2duXPWFHHaJnOEovHXAGzQUl%2FzBocvEbkU%2BstJU3AjUwCqbrDZRVZay0dd7OLQ6NCZTFxvZrhEoOwYukzgYt02XtXs23oNKc8QK8jPBsjSx8K1Xj9V9tPB5eBTptias5DuJnme5GPMo0H0rehhAFIbH1iz5aLEJ2WFAs2YMajNB9b8K9hsdcsufqw3IWvDgGjJl491SPFYv4AkHpfF%2B2OEEmH9hDMp3rnzv686gQ5u2%2FmSKrDsChJ2JSpJUuTnHdHBZrk%2Fwzx%2BpsfpZ4zLVgKx0K6MJOGtdQGOrAB7mHiroFpJKweD4wGdVv4twWKj65wn3ZS715WCDMXnol3ZjKerxKwtTiAS8MYNXREK5yRc6R%2BCtHqgkfO1QIL6y35fyD0iOWFO6ZYrI1IePcEupQe4Ao%2B8i5%2Bz9PydRXrVVCjz1A9o6ZcLX9dN59jVDboJOpZsOpXxuErU9LEKNKGTqVoHfJq%2FpKSdZkltmLsDvSAgBZORP2gpikUUQyl6A0CxzJ3SGD1ZrqXN39XlWY%3D&amp;X-Amz-Algorithm=AWS4-HMAC-SHA256&amp;X-Amz-Date=20260825T084050Z&amp;X-Amz-SignedHeaders=host&amp;X-Amz-Expires=300&amp;X-Amz-Credential=ASIAQ3PHCVTYZDNOMNKH%2F20260825%2Fus-east-1%2Fs3%2Faws4_request&amp;X-Amz-Signature=222054776072dd684a9220ca5e137f808570de2f4b46a74fc1fac36b8741d31e&amp;hash=9077aea1050680fe483f80515f312051569ddf0b7a50ba90c1511bc6ef136d06&amp;host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&amp;pii=S0925753520301922&amp;tid=spdf-2823bc10-4a39-4f2b-8a32-7409653a6f56&amp;sid=6bbf6c2f3a5ff545d91b09f2929365875304gxrqa&amp;type=client&amp;tsoh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&amp;rh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&amp;ua=1b130057565d500d535050&amp;rr=a3094b5109a37504&amp;cc=au]

Footer, K. H. A., White, R. H., Park, J. N., Decker, M. R., Lutnick, A., &amp; Sherman, S. G. (2020). Entry to Sex Trade and Long-Term Vulnerabilities of Female Sex Workers Who Enter the Sex Trade Before the Age of Eighteen. Journal of Urban Health, 97(3), 406–417. https://pmc.ncbi.nlm.nih.gov/articles/PMC7305278/

Huglstad, M., Halvorsen, I. L. I., Jonsson, H., &amp; Nielsen, K. T. (2020). “Some of us actually choose to do this”: The meanings of sex work from the perspective of female sex workers in Denmark. Journal of Occupational Science, 29(1), 1–14. https://doi.org/10.1080/14427591.2020.1830841.

‌Redwood, R. (2018). Myths and Realities of Male Sex Work. Red light labour: Sex work regulation, agency, and resistance, 167-180.[https://books.google.com.au/books?hl=en&amp;lr=&amp;id=8GJoDwAAQBAJ&amp;oi=fnd&amp;pg=PA167&amp;dq=Myths+and+Realities+of+Male+Sex+Work&amp;ots=mQOUglmuLU&amp;sig=XHi0ybe124LABeHMulfVUMeYsxw#v=onepage&amp;q=Myths%20and%20Realities%20of%20Male%20Sex%20Work&amp;f=false]

‌McKeever, N. (2025). Is Sex Work Inherently Gendered? Hypatia, 40(3), 633–652. https://doi.org/10.1017/hyp.2024.56.

McNamara, R. (2016). The trouble with whorephobia. Journal of Religion and Culture, 27(1), 13-34.[https://www.jrc-concordia.ca/wp-content/uploads/2017/01/Journal-of-Religion-and-Culture-Vol-27-no-1.pdf#page=13]}}

==External links==
* [[doi:10.1017/hyp.2024.56|Is Sex Work Inherently Gendered?]] (Cambridge University Press){{ic|Use in references instead}}
* [https://www.jrc-concordia.ca/wp-content/uploads/2017/01/Journal-of-Religion-and-Culture-Vol-27-no-1.pdf#page=13 The Trouble with Whorephobia] (Journal of Religion and Culture){{ic|Use in references instead}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Sex]]
[[Category:Motivation and emotion/Book/Work motivation]]</text>
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    <title>Motivation and emotion/Book/2026/Self-disclosure and emotional intimacy</title>
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&lt;div align=center&gt;[[Motivation and emotion/Book/2025|2026 list of topics]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Stockcouple.jpg|thumb|'''Figure 1'''. Stock image of couple hugging]]

; Picture this...

Jim and Pam have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. 

One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. 

In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before.

Why does this self-disclosure from Jim create this emotional closeness between him and Pam?
{{RoundBoxBottom}}

Key points
* What is self-disclosure?
* Brief overview of some self-disclosure theories (?)
* Brief overview of the importance of emotional closeness

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the relationship between self-disclosure and emotional intimacy?
* What role does reciprocal self-disclosure play in emotional closeness in relationships?
* How can self-disclosure in relationships be fostered?
{{RoundBoxBottom}}

The Overview section (180 to 330 words) (ignore)
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box
Resources (ignore):
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
[[File:Bdspt.model.webp|'''Figure 2.''' Social Penetration Theory (1973) Model.|alt=Figure 2. Social Penetration Theory (1973) Model.|thumb]]

==What is self-disclosure?==

* Define in [[wikipedia:Self-disclosure|self-disclosure]] in further depth
* Different types of self-disclosure
* Self-disclosure theories (subheadings): 
** Social Penetration Theory (Altman &amp; Taylor, 1973) (See figure 2),
** Social Exchange Theory (Thibaut &amp; Kelley, 1959) rewards vs costs of self disclosure

;Quiz
&lt;quiz display=simple&gt;
{Who developed social penetration theory?
|type=&quot;()&quot;}
+ Irwin Altman &amp; Dalmas Taylor
- Carl Rogers &amp; Abraham Maslow
&lt;/quiz&gt;

== Self-disclosure &amp; emotional intimacy ==

*What is [[wikipedia:Emotional_intimacy|emotional intimacy]]? (definition)
*Why we strive for emotional intimacy
*Different types of relationships &amp; emotional intimacy - stranger on the train phenomenon
*How can self-disclosure increase emotional intimacy? How does emotional intimacy build relationships?

==What is reciprocal self-disclosure?==

* Definition
* Norm of reciprocity
* The Dyadic Effect
* Breadth &amp; Depth (SPT, 1973)
* Building trust&lt;br /&gt;

==Promoting self-disclosure==
* How to encourage self-disclosure
* How to respond to self-disclosures
* Boundaries

=== Learning features (ignore) ===
{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action (Can be real or fictional; if real, provide citations)
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1), Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* How does self-disclosure foster emotional closeness in relationships? Signals vulnerability, builds trust, increases understanding, and progresses relationships.
* What is the relationship between self-disclosure and emotional intimacy? Self-disclosure drives emotional intimacy...
* What role does reciprocal self-disclosure play in emotional closeness in relationships? Shared cycle of vulnerability, trust, mutual understanding.
* How can self-disclosure in relationships be fostered? Start slow, reciprocity! Validation &amp; listening. Match pacing and comfortability.
* Take home messages

Notes (ignore):
*The Conclusion is arguably the most important section (150-330 words)
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026)
* [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025)
* [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia)
* [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Altman, I., &amp; Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000

Derlega, V. J., Winstead, B. A., &amp; Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, &amp; J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008

Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129

Greene, K., Derlega, V. J., &amp; Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti &amp; D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' Cambridge University Press. 409-427. https://doi.org/10.1017/CBO9780511606632.023 

Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118]

Tolstedt, B. E., &amp; Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. 

* [https://www.youtube.com/watch?v=Z4bRXozREKU&amp;t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com)
* [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
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&lt;div align=center&gt;[[Motivation and emotion/Book/2025|2026 list of topics]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Stockcouple.jpg|thumb|'''Figure 1'''. Stock image of couple hugging]]

; Picture this...

Jim and Pam have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. 

One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. 

In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before.

Why does this self-disclosure from Jim create this emotional closeness between him and Pam?
{{RoundBoxBottom}}

Key points
* What is self-disclosure?
* Brief overview of some self-disclosure theories (?)
* Brief overview of the importance of emotional closeness

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the relationship between self-disclosure and emotional intimacy?
* What role does reciprocal self-disclosure play in emotional closeness in relationships?
* How can self-disclosure in relationships be fostered?
{{RoundBoxBottom}}

The Overview section (180 to 330 words) (ignore)
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box
Resources (ignore):
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
[[File:Bdspt.model.webp|'''Figure 2.''' Social Penetration Theory (1973) Model.|alt=Figure 2. Social Penetration Theory (1973) Model.|thumb]]

=== What is self-disclosure? ===
* Define in [[wikipedia:Self-disclosure|self-disclosure]] in further depth
* Different types of self-disclosure
* Self-disclosure theories (subheadings): 
** Social Penetration Theory (Altman &amp; Taylor, 1973) (See figure 2),
** Social Exchange Theory (Thibaut &amp; Kelley, 1959) rewards vs costs of self disclosure

;Quiz
&lt;quiz display=simple&gt;
{Who developed social penetration theory?
|type=&quot;()&quot;}
+ Irwin Altman &amp; Dalmas Taylor
- Carl Rogers &amp; Abraham Maslow
&lt;/quiz&gt;

== Self-disclosure &amp; emotional intimacy ==

*What is [[wikipedia:Emotional_intimacy|emotional intimacy]]? (definition)
*Why we strive for emotional intimacy
*Different types of relationships &amp; emotional intimacy - stranger on the train phenomenon
*How can self-disclosure increase emotional intimacy? How does emotional intimacy build relationships?

==What is reciprocal self-disclosure?==

* Definition
* Norm of reciprocity
* The Dyadic Effect
* Breadth &amp; Depth (SPT, 1973)
* Building trust&lt;br /&gt;

==Promoting self-disclosure==
* How to encourage self-disclosure
* How to respond to self-disclosures
* Boundaries

=== Learning features (ignore) ===
{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action (Can be real or fictional; if real, provide citations)
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1), Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* How does self-disclosure foster emotional closeness in relationships? Signals vulnerability, builds trust, increases understanding, and progresses relationships.
* What is the relationship between self-disclosure and emotional intimacy? Self-disclosure drives emotional intimacy...
* What role does reciprocal self-disclosure play in emotional closeness in relationships? Shared cycle of vulnerability, trust, mutual understanding.
* How can self-disclosure in relationships be fostered? Start slow, reciprocity! Validation &amp; listening. Match pacing and comfortability.
* Take home messages

Notes (ignore):
*The Conclusion is arguably the most important section (150-330 words)
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026)
* [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025)
* [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia)
* [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Altman, I., &amp; Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000

Derlega, V. J., Winstead, B. A., &amp; Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, &amp; J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008

Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129

Greene, K., Derlega, V. J., &amp; Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti &amp; D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' Cambridge University Press. 409-427. https://doi.org/10.1017/CBO9780511606632.023 

Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118]

Tolstedt, B. E., &amp; Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. 

* [https://www.youtube.com/watch?v=Z4bRXozREKU&amp;t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com)
* [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Remove link to list of topics (this is available via the breadcrumbs at the top of the page)</comment>
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__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Stockcouple.jpg|thumb|'''Figure 1'''. Stock image of couple hugging]]

; Picture this...

Jim and Pam have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. 

One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. 

In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before.

Why does this self-disclosure from Jim create this emotional closeness between him and Pam?
{{RoundBoxBottom}}

Key points
* What is self-disclosure?
* Brief overview of some self-disclosure theories (?)
* Brief overview of the importance of emotional closeness

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the relationship between self-disclosure and emotional intimacy?
* What role does reciprocal self-disclosure play in emotional closeness in relationships?
* How can self-disclosure in relationships be fostered?
{{RoundBoxBottom}}

The Overview section (180 to 330 words) (ignore)
# '''Scenario''': A short, engaging case study or real-world example in a feature box, with an accompanying image (see above)
# '''Explanation of the problem, issue, or topic''': Briefly explain the problem, why it is important, and outline how psychological science can help
# '''Focus questions''': Unpack the sub-title into focus questions in a feature box
Resources (ignore):
* [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
[[File:Bdspt.model.webp|'''Figure 2.''' Social Penetration Theory (1973) Model.|alt=Figure 2. Social Penetration Theory (1973) Model.|thumb]]

=== What is self-disclosure? ===
* Define in [[wikipedia:Self-disclosure|self-disclosure]] in further depth
* Different types of self-disclosure
* Self-disclosure theories (subheadings): 
** Social Penetration Theory (Altman &amp; Taylor, 1973) (See figure 2),
** Social Exchange Theory (Thibaut &amp; Kelley, 1959) rewards vs costs of self disclosure

;Quiz
&lt;quiz display=simple&gt;
{Who developed social penetration theory?
|type=&quot;()&quot;}
+ Irwin Altman &amp; Dalmas Taylor
- Carl Rogers &amp; Abraham Maslow
&lt;/quiz&gt;

== Self-disclosure &amp; emotional intimacy ==

*What is [[wikipedia:Emotional_intimacy|emotional intimacy]]? (definition)
*Why we strive for emotional intimacy
*Different types of relationships &amp; emotional intimacy - stranger on the train phenomenon
*How can self-disclosure increase emotional intimacy? How does emotional intimacy build relationships?

==What is reciprocal self-disclosure?==

* Definition
* Norm of reciprocity
* The Dyadic Effect
* Breadth &amp; Depth (SPT, 1973)
* Building trust&lt;br /&gt;

==Promoting self-disclosure==
* How to encourage self-disclosure
* How to respond to self-disclosures
* Boundaries

=== Learning features (ignore) ===
{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action (Can be real or fictional; if real, provide citations)
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1), Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* How does self-disclosure foster emotional closeness in relationships? Signals vulnerability, builds trust, increases understanding, and progresses relationships.
* What is the relationship between self-disclosure and emotional intimacy? Self-disclosure drives emotional intimacy...
* What role does reciprocal self-disclosure play in emotional closeness in relationships? Shared cycle of vulnerability, trust, mutual understanding.
* How can self-disclosure in relationships be fostered? Start slow, reciprocity! Validation &amp; listening. Match pacing and comfortability.
* Take home messages

Notes (ignore):
*The Conclusion is arguably the most important section (150-330 words)
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026)
* [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025)
* [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia)
* [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Altman, I., &amp; Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000

Derlega, V. J., Winstead, B. A., &amp; Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, &amp; J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008

Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129

Greene, K., Derlega, V. J., &amp; Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti &amp; D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' Cambridge University Press. 409-427. https://doi.org/10.1017/CBO9780511606632.023 

Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118]

Tolstedt, B. E., &amp; Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. 

* [https://www.youtube.com/watch?v=Z4bRXozREKU&amp;t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com)
* [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relationships]]</text>
      <sha1>2qavxyxjfylbj4aitcbfxq3eione72q</sha1>
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  <page>
    <title>Motivation and emotion/Book/2026/Moral disgust and jury decision-making</title>
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      <contributor>
        <username>Yellowvines</username>
        <id>3106353</id>
      </contributor>
      <comment>/* Moral Disgust */</comment>
      <origin>2830643</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20660" sha1="pp1rhw8ei3z25rtfliyeslc7wzzlyz9" xml:space="preserve">{{title|Moral Disgust and Juror Decision-Making:

How does moral disgust impact jurors judgements of guilt, blame and punishment?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}[[File:Trial by Jury - Chaos in the Courtroom.png | thumb | left | 350px | Figure 1: A jury is a group of sworn individuals who determine a defendant's guilt. ]]

Consider sitting on a jury. The defendant appears composed, well-dressed, and speaks clearly. Someone like this couldn't have done anything bad — that's your instinct.

Then you hear the charge: embezzlement. Over three years, they quietly transferred company funds to a personal account. Thousands of dollars were spent, and real people were harmed. You are angry. It's wrong. However, it does not make you feel sick.

Consider the same composed defendant in the same courtroom, but this time the charge is posing as a charity volunteer to gain the trust of dying hospice patients before stealing the donations intended for their care. The financial loss is the same as the embezzlement case. Same amount, same number of victims.

But something else arises in you. Not just anger, but more of a revulsion, as if something was contaminated. This doesn't simply feel like a crime. It has an unclean feeling to it.

The same defendant. Same harm. So, why does one case make you feel cold and angry, while the other makes your skin crawl? And, aside from the actual harm done, could that visceral reaction be influencing how harshly you believe they should be punished?


{{RoundBoxBottom}}

==== The Topic ====
It is the law that [https://www.monash.edu/law/news-and-events/news/2020/juries-why-do-we-actually-need-them-and-can-they-get-it-wrong jurors] must base their verdicts on the facts of a case. But moral disgust, an emotional gut reaction, seems to influence how jurors assess guilt, blame and punishment, regardless of the strength of the evidence presented. A disgusted juror might unconsciously become more certain of guilt, assign more blame and suggest harsher punishment, regardless of what the facts actually establish. If disgust always biases judgements and sentences in one direction or the other, no matter how strong a case is, the implications are profound: wrongful convictions, disproportionate sentencing, and differences in the punishment of similar crimes depending on how “disgusting” the evidence seems, rather than the true seriousness of the crime. It also raises questions for current legal practices: courts commonly admit gruesome evidence under the assumption that its evidentiary value outweighs its emotional impact, an assumption this body of research directly undermines. Psychological science provides the legal system with something it cannot produce itself: controlled empirical evidence about the nature and causes of this bias. It shows not only that jurors appear to be affected by emotion but also the processes (appraisal patterns, purity vs. harm, violations, and individual differences such as private body consciousness) that determine when and for whom the effect is strongest. 


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'''Focus questions'''

*If disgust can make moral judgements more severe, even when it’s incidental to the act being judged, what does that say about the reliability of human moral reasoning?
*Can moral disgust be used by the legal system?
*How might a juror's own sensitivity to physiological sensations of disgust impact how vulnerable they are to disgust-driven bias?
*How can psychological theories be used to mitigate risks of bias in the legal system?


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==Headings==
* [[#Overview|Overview]]
* What is Moral Disgust?
* Psychological Science in the Legal System
* Disgust as a Legally Relevant Emotion
* Moral Disgust and Juror Judgements
* Moderating Factors
* Practical Implications
* [[#Conclusion|Conclusion]]
* See also
* References
* External links

== What is Moral Disgust? ==

=== Disgust ===
[[w:Disgust|Disgust]] is considered one of the six universal emotions that humans experience and can be visually recognised on a person's face. The emotion of disgust is theorised to have originated from a biological protective mechanism to protect an organism from harm that may be brought on by food or poison. Disgust has now evolved to become an emotion/behaviour directed at anything that may potentially cause harm or disease. For example, 

* body products 
* animals 
* hygiene 
* body violations, death or disease
Paul Ekman proposed six basic emotions in his literature based on the theories that these emotions, across research all over the world, concluded in agreement on six emotions: happiness, surprise, fear, anger, sadness and disgust. Therefore, he proposed that the universality of these findings and the facial expression alone could conclude that there are 6 basic emotions (Ekman, 1992)

=== Moral Disgust ===
Moral disgust refers to the emotional response a person feels when their social or cultural norms are transgressed, therefore resulting in a reaction. Moral disgust is complex, and there is varying evidence for and against its genuine purpose as an emotion. Researchers argue that moral disgust is a metaphorical extension of physical disgust; shared facial expression, language, etc. Other researchers argue that it's functionally unique to traditional, physical disgust. It does not reliably trigger the same physiological processes: facial muscle activation, nausea, etc. (Chapman et al., 2012). 

Moral disgust is triggered by acts that are supposedly a violation of an individual's perceived purity or dignity. For example, betraying someone's trust (cheating on a partner) or exploiting a vulnerable person (as mentioned in the scenario at the start). 

Rozin et al. (1999) proposed a theory that moral disgust specifically clusters around violations of a person's divinity or purity. This is different to anger which generally clusters around autonomy violations, also opposed to contempt which clusters around community violations. 

This distinction is vital when discussing emotion for jurors. If moral disgust behaves differently from physical disgust, it may influence judgement in the courtroom through a different pathway than a simple reaction to physically disgusting evidence. &lt;quiz display=&quot;simple&quot;&gt;

{Moral disgust and physical disgust are the same emotion:
|type=&quot;()&quot;}
- True
+ False


&lt;/quiz&gt;

== Psychological Science in the Legal System ==

* [[Motivation and emotion/Book/2019/Criminal record stigma and emotion#Relevant theories|What psychological theories are present in the legal system]]
**social identity theory 
***a theory that posits that whole organisations and groups that possess the same cultural and social norms will all have similar self-concepts (Bornewasser &amp; Bober, 1987).
***The theory predicts that behaviours from this group will often be similar due to their similar self-concept. 
**[[w:Appraisal_theory|appraisal theory of emotion]]
***appraisal theory of emotion is theory that specific emotions are extracted from evaluations of events which in turn impact a reaction which may differ depending on certain people. 
**evolutionary theory of emotion
***this theory describes our emotions and reactions being biological reactions to stimuli. 
****for example, disgust has come from protecting oneself from poisons. 

== Disgust as a Legally Relevant Emotion ==

* Disgust should not be seen as legally relevant, as that would be a personal bias. 
* How can you create a courtroom that is free from bias if you do not take into account the impact of disgust? 
* Maybe disgust shouldn't be seen as an impacting emotion but rather a factor that needs to be considered when a jury is shown pictures, footage, and witness statements in court. 

=== Arguments For ===

* Patrick Devlin's belief that we should allow some degree of the shared morality view of society through emotions 
** Within society, when crimes are viewed as &quot;disgusting&quot; or cause widespread intolerance, there is an importance to the intervention of the legal system (Lacey, 2023). 
* Informing the &quot;Reasonable Person&quot; Standard 
** This is a theory that jurors will judge acts (particularly those involving negligence or abuse) based on what they believe they would have done in that situation. Or more accurately to the theory, what would a reasonably prudent person (RPP) have done (Alicke &amp; Weigel, 2021)? 

=== Arguments Against ===

* [[wikipedia:Harm_principle|harm principle limitation]] 
** Developed by John Stuart Mills, the harm principle limitation states the following: 
** &quot;The only purpose for which power can be rightfully exercised over any member of a civilised community, against his will, is to prevent harm to others.&quot; [[wikisource:On_Liberty/Chapter_1|(Mill, 1869).]] 
** This principle bars people within the legal system from criminalising an action because it makes most people feel moral disgust. 
* a tool for discrimination
** Historically, moral disgust has been a tool against those being charged with crimes of homosexuality and interracial marriages. 

== Moral Disgust and Juror Judgements ==
A jury is a group of 12 individuals that are carefully chosen to determine the guilt of an offender. They are picked to be impartial, unbiased opinions that can aid in the determining of a person's guilt. Jurors are asked to make judgements based on the evidence presented to them in court. However, emotions play a large, unsuspecting role in determining the guilt of a person. Such is the reason why, in high-stakes cases, jurors are not allowed contact with the outside world or media during trials. Moral disgust in particular seems to have a large impact on the way that jurors perceive and shape jurors' judgements. Different emotions elicit different reactions based on the emotion and a person's appraisal patterns.  

=== Emotional Response Patterns ===
Fishbein &amp; Ajzen's (1975) expectancy-value model states that a person's given attitude towards an object is a direct function of the value that a person attaches to that object's attributes or outcomes (Desteno et al., 2004).  

Emotional appraisal patterns are the specific mental evaluations that an individual's brain makes about an event to determine how they feel about a situation. Older research has treated negative emotions as one category. But it is more complicated than that. Not all emotions push judgement in the same direction, which creates moral disgust and other emotions in their own category. 

INSERT TABLE HERE W/ EVIDENCE  

== Moderating Factors ==

* private body consciousness 
** people who are more prone to experience their private bodily sensations show stronger disgust-driven bias 
*** An experiment conducted by Schnall et al. (2008) revealed that people who were more aware of their Private Body Consciousness (PBC) were more prone to feelings of disgust and expressing these feelings outwards to the situation at hand. 
* type of violation 
** purity vs harm

== Practical Implications ==

* evidentiary admissibility 
** courts allow for the admission of gruesome photographic evidence but findings suggest this may be wrong 
*** research shows that people view crimes harsher if exposed to gruesome photos 
*** Additionally, Bright &amp; Goodman-Delahunty's (2006) research discusses a person's use of their inner emotional states as a way to reason with information they are being given.
*** In their affective influences model, affect is a judgement-simplifying heuristic device. People generally consult their affective state, meaning they consult their inner emotion to infer a judgement before looking at the facts. 
* jury selection
* how evidence is presented 

*

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

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{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
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A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
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Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
==Conclusion==
* What does this mean going further
* What would future research benefit from in this specific section?
*Moral disgust does not come with jurors' judgements, but rather it can measurably shift guilt, certainty, blame attribution and punishment severity.
*This may be stronger for certain people; it cannot just be a blanket claim for all jurors. 
*The idea that jurors decide on the basis of facts alone is not psychologically sustainable. The practice of law needs to consider this, rather than seeing emotions as a bias. 
* The Conclusion is arguably the most important section
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}}

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Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
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* [[w:Self determination theory|Self determination theory]] (Wikipedia)

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}}

==References==
{{Hanging indent|1=

Alicke, M. D., &amp; Weigel, S. H. (2021). The reasonable person standard: Psychological and legal perspectives. Annual Review of Law and Social Science, 17(1), 123–138.

Bornewasser, M., &amp; Bober, J. (1987). Individual, social group and intergroup behaviour. Some conceptual remarks on the social identity theory. European Journal of Social Psychology, 17(3), 267–276. https://doi-org.ezproxy.canberra.edu.au/10.1002/ejsp.2420170303

Bright, D. A., &amp; Goodman-Delahunty, J. (2006). Gruesome evidence and emotion: Anger, blame, and jury decision-making. Law and Human Behavior, 30(2), 183–202. https://doi.org/10.1007/s10979-006-9027-y

Chapman, H. A., &amp; Anderson, A. K. (2012). Understanding disgust. Annals of the New York Academy of Sciences, 1251(1), 62–76. https://doi.org/10.1111/j.1749-6632.2011.06369.x 

DeSteno, D., Petty, R. E., Rucker, D. D., Wegener, D. T., &amp; Braverman, J. (2004). Discrete emotions and persuasion: The role of emotion-induced expectancies. Journal of Personality and Social Psychology, 86(1), 43–56. https://doi.org/10.1037/0022-3514.86.1.43

Ekman, P. (1992). Are there basic emotions? Psychological Review, 99(3), 550–553. https://doi.org/10.1037/0033-295X.99.3.550

Fishbein, M., &amp; Ajzen, I. (1975). Belief, attitude, intention, and behavior: an introduction to theory and research. Addison-Wesley Pub. Co.

Lacey, N. (2023). Patrick Devlin, The enforcement of morals (1965). In [Book title] (1st ed.). Routledge. https://doi.org/10.4324/9781003193982-5

Schnall, S., Haidt, J., Clore, G. L., &amp; Jordan, A. H. (2008). Disgust as embodied moral judgment. Personality and Social Psychology Bulletin, 34(8), 1096–1109. https://doi.org/10.1177/0146167208317771

{{Hanging indent|1=}}

}}

{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
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}}

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Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
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* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
* [https://www.youtube.com/watch?v=u-TmKo75gJI How Disgust Shapes our Thoughts on Moral Wrong &amp; the Political Right] (David Pizzaro)

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}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Disgust]]
[[Category:Motivation and emotion/Book/Morality]]
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    <title>Motivation and emotion/Book/2026/Extended process model of emotion regulation</title>
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      <text bytes="10551" sha1="3jh2aubhdv8c7utmfdhbpjs53upsfqn" xml:space="preserve">{{title|Extended process model of emotion regulation:&lt;br&gt;What is the extended process model and how does it explain the regulation of emotions in different contexts?}}
__TOC__
==Overview==
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[[File:School level test of Wikischolar 2026 at Ispahani Public School &amp; College 08.jpg|thumb|Figure 1: Stressful exam]]

''' Case Study - Exam stressors'''

Michael is halfway through a university exam, in a small classroom filled with 30 other examinees. Suddenly, he realises that he misread a question and wasted the past 15 minutes. His heat rate spikes, thoughts start racing and begins to panic. Becoming overwhelmed at the sudden onset of negativity, he felt the urge to give up.

Normally, Michael copes with [[W: Psychological stress | stress]] and [[anxiety]] through exercise as he finds this is the most consistent modulator for his emotions. But in an exam room, this strategy becomes moot.

Michael stops, realises that the current negativity isn't helping his situation, and identifies that panic is only hindering his performance and that emotional regulation is needed if he wants to do well. He starts to tell himself &quot;I can still get partial marks for that question&quot;. Realising that the positive self-talk is helping he continues &quot;On this next question I'll make sure to focus on all facet's of what the question asks&quot;. By the end of the next question Michael once again felt a sense of confidence in his ability to pass the exam.{{RoundBoxBottom}}

Emotion regulation refers to the processes through which individuals monitor, evaluate and modify their emotional responses. Contemporary psychological science emphasises that regulation is not a singular act but a dynamic, context-sensitive system continuously unfolding over time (Bonanno &amp; Burton, 2013). Gross's (2015) Extended Process Model (EPM) captures this complexity by organising regulation into three iterative stages: identification, selection and implementation. Each governed by higher-order valuation system that determine whether, when, and how emotions should be regulated. This framework highlights that regulatory success depends not only on the strategy chosen but on the fit between regulatory goals, contextual affordances, and the individual's cognitive resources.

Emotion regulation is central to mental health. [[Motivation and emotion/Book/2026/Emotion dysregulation|Dysregulation]] predicts a wide range of difficulties, including the strong associated between emotional dysregulation and anxiety in autistic youth (Connor et al., 2020). Meta-analytic evidence further demonstrates that habitual reappraisal is linked to better wellbeing, whereas suppression is often linked to poorer outcomes, though these patterns seem to vary across contexts (Chen et al., 2025). These findings underscore the need to understand ''why'' regulation works differently across individuals and situations.

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'''Focus questions'''

* How do cultural [[W: social norms | norms]] shape the valuation of regulatory goals within the EPM?
* What role does [[W: cognitive flexibility | cognitive flexibility]] play in the EPM’s ability to explain individual differences in regulatory effectiveness?
* How does the EPM explain the strong predictive link between [[Motivation and emotion/Book/2026/Emotion dysregulation | emotional dysregulation]] and anxiety in autistic youth?
* In what types of situations does switching to distraction improve regulatory outcomes, and why might switching to [[W: cognitive appraisal |reappraisal ]] impair them?
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*

== The Extended Process Model of Emotional Regulation ==

* Emotion regulation as a higher‑order valuation system modifying first‑order emotional responses (Gross, 2015).
* Three‑stage regulatory cycle: '''identification''', '''selection''', '''implementation''' (Gross, 2015).
* Emphasis on temporal dynamics, regulation unfolds iteratively and adjusts based on feedback.
* Situated cognition: regulatory success depends on contextual affordances, not inherent strategy value (Gross, 2015).
* Sets the foundation for exploring cultural norms, cognitive flexibility, developmental trajectories, and strategy effectiveness.


&lt;quiz display=&quot;simple&quot;&gt;
{Which of these is not a stage in the regulatory cycle?|type =&quot;()&quot;}

+ Yelling 
- Identification
- Selection
- Implementation

{According to EPM, After Selecting One Strategy to Use, Emotional Regulation is Done:}

- True
+ False

{Improvements will be made to this quiz}

+ True
- False
&lt;/quiz&gt;

==The Effects of Culture on Emotional Regulation==

* Cultural norms shape valuation systems that determine regulatory goals and strategy desirability (Chen et al., 2025).
* Reappraisal linked to better mental health across cultures, but strength of association varies with cultural dimensions such as uncertainty avoidance and long‑term orientation (Chen et al., 2025).
* Suppression more adaptive in cultures emphasising restraint or collectivist norms (Chen et al., 2025).
* Cultural variation foreshadows how the EPM’s identification and selection stages are culturally embedded.
* Leads into the question: ''How do cultural norms shape the valuation of regulatory goals within the EPM?''

== Cognitive Flexibility in Emotional Regulation ==

* Regulatory flexibility requires context sensitivity, a diverse repertoire, and responsiveness to feedback (Bonanno &amp; Burton, 2013).
* Cognitive flexibility supports adaptive switching when strategies prove ineffective.
* Physiological markers (e.g., corrugator EMG, LPP, SPN) predict switching decisions, indicating cognitive‑affective monitoring processes (Adamczyk et al., 2024).
* Flexibility explains individual differences in regulatory effectiveness beyond strategy type.
* Leads into the question: ''What role does cognitive flexibility play in the EPM’s ability to explain individual differences in regulatory effectiveness?''

== Children and Emotional Regulation ==

* Children demonstrate early regulatory flexibility, including spontaneous strategy use and switching (Wong et al., 2019).
* Age increases likelihood of cognitively demanding strategies such as reappraisal (Wong et al., 2019).
* Negative emotion intensity prompts greater use of disengagement strategies like distraction (Wong et al., 2019).
* Developmental patterns highlight how identification and selection processes mature across childhood.
* Provides developmental context for understanding dysregulation in clinical groups.
* Dysregulation strongly predicts anxiety in autistic youth, independent of core symptoms (Conner et al., 2020).
* Leads into the question: ''how does EPM explain the strong predictive link between emotional dysregulation and anxiety in autistic youth?''

== What Emotional Regulation Strategies are Most Effective ==

* Reappraisal generally linked to better mental health outcomes; suppression linked to poorer outcomes, though context moderates these effects (Chen et al., 2025).
* Distraction and reappraisal reduce negative affect but operate through distinct attentional and cognitive mechanisms (Strauss et al., 2016).
* Switching to distraction improves neural downregulation; switching to reappraisal can impair it under high‑intensity conditions (Adamczyk et al., 2024).
* Leads into the question: ''In what situations does switching to distraction improve outcomes, and why might switching to reappraisal impair them?''

==Conclusion==
* Emotional regulation is a dunamic, multi-stage system invovling '''identification, selection,''' and '''implementation''' of regulatory strategies (Gross, 2015)
* The EPM conceptualises regulation as a higher-order valuation process, continuously monitoring emotional states and adjusting strategies based on contextual feedback (Gross, 2015)
* Regulation is not inherently adaptive or maladaptive; it's effectiveness depends on contextual affordances, cognitive resources and situational demands (Gross, 2015)
* EPM provides a mechanistic framework explaining why individuals differ in regulatory success, integrating cognitive flexibility, context sensitivity and strategy repertoire (Bonanno &amp; Burton, 2013). 
* EPM offers a structure for understanding cross-cultural variation in regulatory goals and strategy valuation (Chen et al., 2025)

{{tip|Take Home Messages:
* Effective emotion regulation is flexible, not fixed; adaptability is more important than any single strategy.
* Cultural norms, cognitive flexibility, and developmental factors shape how regulation unfolds
* Dysregulation is a powerful predictor of mental health difficulties, reinforcing the need for models like the EPM to guide intervention
* Strategy effectiveness is context-dependent
* The EPM provides a cohesive, empirically grounded framework for understanding the complexity of human emotional life
}}
*

== See Also: ==
[https://www.youtube.com/watch?v=gAMbkJk6gnE What are emotions? and why do we feel them?]

[https://www.youtube.com/watch?v=Yd6hR1qCfSM Emotion Regulation with James J. Gross, PhD]

==References==
{{Hanging indent|1=

Adamczyk, A. K., et al. (2024). Emotion regulation flexibility: EEG, EMG predictors and consequences of switching. Psychophysiology, 61(7). https://doi.org/10.1111/psyp.14646

Bonanno, G. A., &amp; Burton, C. L. (2013). Regulatory flexibility: An individual differences perspective on coping and emotion regulation. Perspectives on Psychological Science, 8(6), 591–612.

Chen, X., Cai, Q., Omari, D., Sanghvi, D. E., Lyu, S., &amp; Bonanno, G. A. (2025). Emotion regulation and mental health across cultures: A systematic review and meta-analysis. Nature Human Behaviour. https://doi.org/10.1038/s41562-025-02168-8 (doi.org in Bing)

Conner, C. M., et al. (2020). The role of emotion regulation and core autism symptoms in the experience of anxiety in autism. Autism, 24(4), 931–940.

Gross, J. J. (2015). The extended process model of emotion regulation: Elaborations, applications, and future directions. Psychological Inquiry, 26(1), 130–137.

Strauss, G. P., Ossenfort, K. L., &amp; Whearty, K. M. (2016). Reappraisal and distraction emotion regulation strategies are associated with distinct patterns of visual attention and differing levels of cognitive demand. PLoS ONE, 11(11), e0162290.

Wong, P. S., et al. (2019). Emotion regulation strategies in children: Developmental patterns and cognitive mechanisms. Journal of Experimental Child Psychology, 183, 1–18.

}}

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==Overview==
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'''The Film &quot;Inside Out&quot; and the challenge of measuring emotion'''

[[File:Inside-out-5492d0c4e3912.jpg|110px|thumb|'''Figure 1. '''Visual representation of Riley’s emotional experiences in Inside Out.]]

Imagine watching Pixar's ''Inside Out''. As Riley struggles to adjust to moving to a new city, viewers experience her changing emotional world alongside her. As illustrated in Figure 1, her emotions are represented as distinct characters, making her internal experiences visible to the audience. Yet, despite watching the same film, two viewers may experience it differently. One viewer might feel sadness as Riley misses her old home, while another may feel hopeful as she begins to adapt.
{{RoundBoxBottom}}
This raises an important question for businesses and researchers alike: ''How can we actually know what consumers are feeling?''
If a company wanted to evaluate audience reactions to Inside Out, would it rely on questionnaires asking viewers to self report, should it analyse facial expressions, measure heart rate, or track physiological arousal? How do we approach the two viewers reporting differences in emotion despite watching the same film?

The challenge is that emotions are not directly observable, unlike the film Inside Out. Instead, researchers must infer emotional experiences through psychological, behavioural, physiological responses. These emotional experiences are important to businesses because they can influence consumer attitudes, product evaluations, satisfaction and purchasing decisions. 

This chapter explores what consumer emotion is, how it can be measured, why consumers may respond differently to the same stimulus, and how individual differences and consumer segmentation can be considered when interpreting emotional responses.{{RoundBoxTop|theme=2}}
'''Focus questions'''
* What is consumer emotion?
* What methods are used to measure consumer emotion?
* How do individual differences influence consumers' emotional responses and their measurement?
* How can consumer emotion measurement and segmentation inform business decisions?
{{RoundBoxBottom}}

==Understanding consumer emotion==
The influence of emotions on judgements, evaluations and decision-making has long been recognised within psychology and consumer behaviour. Early research established that emotions can influence how consumers respond to products and consumption experiences, while more recent research has developed an understanding of how specific emotional states can produce different effects on consumer decision making. Emotions can shape how consumers evaluate products, brands and experiences, while consumers may also seek to regulate their emotions throughout the consumption process.

Understanding these emotional processes is therefore important for examining how and why consumers respond to products and marketing stimuli in different ways. Before considering how these responses can be measured, it is necessary to first understand the nature and dimensions of consumer emotion. This section highlights the early research, key terms and concepts, and further progresses to the current understanding of consumer emotion to date.

=== '''Dimensions of emotional experience''' ===

* Emotion is not a single, uniform psychological state; researchers have proposed small sub-sets of dimensions to identify and classify &quot;basic&quot; emotions (Stein &amp; Trabasso, 1992).
* Psychologist [[w:Paul_Ekman|Paul Ekman]] identified the most widely recognised set of [https://www.google.com/search?q=6+basic+emotions&amp;rlz=1C5CHFA_enAU1144AU1144&amp;oq=6+basic+emotions&amp;gs_lcrp=EgZjaHJvbWUqDQgAEAAYkQIYgAQYigUyDQgAEAAYkQIYgAQYigUyDAgBEAAYFBiHAhiABDIHCAIQABiABDIHCAMQABiABDIHCAQQABiABDIHCAUQABiABDIGCAYQRRg8MgYIBxBFGDzSAQgyMDQ1ajBqNKgCALACAQ&amp;sourceid=chrome&amp;ie=UTF-8 six basic human emotions] (happiness, sadness, surprised, fear, anger and disgust) in the 1960s, grounded in Darwin's theory (Ekman, Dalgleish &amp; Power, 1999).

*Two major approaches are '''dimensional emotion models,''' which asserts feelings as continuous points on a coordinate grid rather than as seperate, isolated categories. And second, '''categorical emotion models (also referred to as discrete models),''' a framework that classifies human feelings into discrete, seperate and biologically innate basic categories. (PS &amp; Mahalakshmi, 2017).
**Ekman's work denotes a less-complex '''categorical model''' of conceptualising emotional experiences.
*Consumer research has historically used dimensional approaches such as [[w:Valence_(psychology)|valence]] and [[w:Arousal|arousal]] (see key terms and figure 2), however, more recent research suggests that examining specific, discrete emotions can provide additional information about consumer judgements and behaviour (Harmon-Jones, Harmon-Jones &amp; Summerell, 2017).


[[File:Circumplex model of emotion.svg|center|thumb|400x400px|'''Figure 2.''' Russell's Circumplex Model of emotion, a dimensional approach to classifying emotional experience (Russell, 1980).]]


'''Table 1'''

The evolution of dimensional approaches to emotions 
{| class=&quot;wikitable&quot;
|+
!Model
!Properties
!Author and Year
|-
|&lt;u&gt;Pleasure-Arousal-Dominance (PAD) emotional state model&lt;/u&gt; 
|Mehrabian and Russell proposed that emotional responses to environments could be described using three dimensions: '''pleasure, arousal and dominance.'''
|(Mehrabian &amp; Russell, 1974).
|-
|&lt;u&gt;Russell's Circumplex Model&lt;/u&gt;
|Russell (1980) proposed that affective experiences can be represented within a circular, two-dimensional space. The model is primarily organised around '''pleasure (valence)''' and '''arousal.'''
|(Russell, 1980).
|-
|&lt;u&gt;Thayer Model of mood and emotion&lt;/u&gt;
|Thayer's model describes how activated and tense a person feels on the basis of two dimensions (energetic arousal and tense arousal).
|(Thayer, 1989).
|-
|&lt;u&gt;The Circumplex Model of affect&lt;/u&gt;
|Russell and Barrett (1999) argued that '''“'''emotion” is too broad to be treated as one scientific category, meaning that different things referred to as emotion may need to be distinguished. A prototypical emotional episode is more complex and involves a sequence of processes, including an antecedent, appraisal, physiological and cognitive changes, behavioural responses and self-categorisation.
|(Russell &amp; Barrett, 1999).
|}
Evident in Table 1, the shift from 1974 to 1999 alone denotes the research shift from a primarily dimension and continuum focused approach, to an integration of both dimensional and categorical perspectives. The circumplex model of affect notes that &quot;emotion is too broad&quot; and suggested a shift to a categorisation approach. Furthermore, the 2000s welcomed a prevalence of categorical approaches (Kranzbühler et al., 2020).  

Early dimensional models provided a useful way of representing emotional experience along continuous dimensions such as valence and arousal. However, contemporary research suggests that these broad dimensions may not capture the full psychological and behavioural significance of specific emotions. Rather than replacing dimensional approaches, contemporary research increasingly recognises the value of considering both dimensional and discrete perspectives. This broader understanding of emotional experience provides a foundation for considering how emotions influence consumers' responses to products, brands and consumption experiences. 
{{RoundBoxTop|theme=2}}
'''(1) Dimensions of emotional experience - Take home knowledge:'''

Consumer emotions can be understood using both dimensional approaches, which describe emotions according to their fit with broad dimensions, such as valence and arousal, and categorical approaches, which distinguish specific emotions such as happiness, anger and fear. Contemporary consumer research recognises that both perspectives provide valuable insights into emotional experiences.
{{RoundBoxBottom}}
=== Emotion and consumer behaviour ===

* Emotions impact consumer decision-making through [[w:Cognitive_appraisal|cognitive appraisals]] (Achar et al., 2016).
*Positive and negative emotions can influence evaluations differently, but the specific emotion experienced may also matter e.g., according to the Circumplex model, anger and fear are both negative/high-arousal emotions, but can produce different judgements and decisions, supporting the need for a categorical model influence.
*Lerner and Keltner (2001) demonstrated this distinction using the '''Appraisal-Tendency framework,''' two emotions with the same negative valence produce inverse effects on decision-making.

=== The application of consumer emotion ===

* Understanding consumer responses &gt; How do emotional responses provide insight beyond satisfaction (Zeelenberg &amp; Pieters, 2004).
* Informing Business decisions &gt; product development, advertising strategies, branding and consumer experience
* Businesses want to apply this information accordingly, we must find reliable ways to measure responses.

== Measuring consumer emotion ==

Emotions involve subjective experiences as well as observable behaviour and physiological responses, researchers use multiple approaches to capture different facets of emotion which will be discussed in this section.

===== Self-report measures =====
Self-report measures ask consumers to consciously report their emotional experiences using emotion scales or questionnaires.

'''Table 2'''

Summary of self-report tools used to measure consumer emotion. 
{| class=&quot;wikitable&quot;
|+
!Tool
!Description
!Advantage
!Limitation
|-
|
|
|
|
|-
|
|
|
|
|-
|
|
|
|}

*Self-report measures capture subjective emotional experiences, that may not be observable through '''behavioural''' or '''physiological measures''' (Chamberlain &amp; Broderick, 2007).

*Such methods are used to assess emotional responses to advertisements, products and consumption experiences, but may be influenced by recall and response biases. 

====== '''Behavioural and physiological measures''' ======

* Emotional responses can be inferred from observable behaviour and physiological changes, such as facial expressions, heart rate and skin conductance. The '''James-Lange theory of emotion (1884)''' (made early note of the relationship between physiological changes and perceived emotion (James, 2002; Ningjian, 2024).

* Considerations &amp; limitations
'''Concluding para:''' Address limitations, analyse current measurements (shopping cites etc..), link to early theories.

== Individual differences in consumer emotion ==

Although measurement methods can provide insight into consumer emotion, emotional responses should not be assumed to be consistent across consumers. Individuals may experience and express different emotions in response to the same stimuli due to differences in physiological or innate characteristics, experiences and appraisal processes. Thereby, understanding these differences  is important when interpreting consumer emotion measurements as a &quot;whole&quot; concept. This section will dissect individual differences, personality and emotion, and further the implications this has on standard emotion measurement.

=== '''Individual Differences''' ===

* Individualistic vs collectivist cultures affect the upbringing of individuals, leading to different conceptualisations of emotions, generalising standard tests/measures to large cross-cultural groups can be inapplicable (Matsumoto &amp; Hwang, 2012).
* As touched on previously, Ekman's six &quot;basic&quot; emotions is the most widely recognised emotion theory globally (Ekman, Dalgleish &amp; Power, 1999).
*Research by Xie, Bagozzi &amp; Grønhaug (2015) identified the role of moral emotions and their influence on consumers responses to corporate environmental actions. The study additionally highlighted how individual characteristics shape these emotional reactions.Findings showed that corporate non-green actions triggered negative emotions such as contempt, anger and disgust, particularly among consumers with stronger social justice values, empathy, moral identity and self-concept, which subsequently led to negative word-of-mouth, complaints and boycotting.In contrast, corporate green actions elicited gratitude, particularly among more empathetic consumers, which encouraged positive responses such as positive word-of-mouth, resistance to negative information, stronger company identification and investment. 

=== '''Personality and emotion''' ===

* Western individualistic cultures have permeated the understanding of personality and emotion, this is an important consideration when applying concepts broadly or administering psychological measures among large groups.
* Consumption-related emotions are often operationalised as broad dimensions, however, research suggests these dimensions (valence and arousal) have been shown to be influenced by specific personality traits and further, influence customer satisfaction.  There is a growing importance of categorical emotional measurement over dimensional (Faullant, Matzler &amp; Mooradian, 2011).
** Study looks at categorical emotions linked to personality (Joy = extraversion)(Fear = neuroticism). 
** Implications on consumer emotion measurement

== Consumer segmentation and emotion measurement ==
Recognising individual differences in emotional responses has implications beyond measurement, as emotional information can also help businesses identify meaningful consumer groups. When emotion measurement is integrated with consumer segmentation, businesses can examine whether different groups of people respond to products, brands or marketing stimuli differently.

=== Emotional Segmentation ===

*Consumer segmentation identifies groups with shared characteristics, while emotional segmentation considers different emotional needs, responses and motivations.
*Consumers can associate different emotions with products and brands, providing a basis for identifying distinct consumer groups (Bigné &amp; Andreu, 2004).
*Businesses can identify segments with different emotional needs to marketing stimuli 
**Case study (brand) example.

=== Utilising emotion data to inform marketing ===

* Emotional responses provide information that can support targeted positioning and communication strategies.
* Specific emotions can influence consumer judgements and behaviours differently.
* Emotion data can inform advertising, product development, branding and customer experience decisions.

* Limitations and ethical considerations

{{RoundBoxTop|theme=8}}
'''(2) Knowledge check quiz'''
&lt;quiz display=simple&gt;
{'''All consumers will experience the same emotion when exposed to the same product or marketing stimulus''':
|type=&quot;()&quot;}
- True
+ False

{'''Emotions must be inferred through measurable responses''':
|type=&quot;()&quot;}
+ True
- False
&lt;/quiz&gt;
{{RoundBoxBottom}}

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

==Conclusion==
* Consumer emotion is complex and cannot be directly observed, so it must be inferred through measurable responses.
* Self-report, behavioural and physiological measures capture different aspects of emotional experience.
* Psychological theories, including dimensional and categorical models, guide how emotion is measured.
* Individual differences such as personality, culture and values determine why consumers may respond differently to the same stimulus.
* Combining measurement approaches and considering individual differences can provide more meaningful consumer insights.
* Emotion measurement can support consumer segmentation and business decisions across many business activities.
* Effective consumer emotion measurement relies on an integration of multiple methods and contextual interpretation.
==See also==
* [[w:Consumer behaviour|Consumer behaviour]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Emotional buying|Emotional Buying]] (Book chapter, 2021)
* [[Motivation and emotion/Book/2019/Personality and emotion|Personality and Emotion]] (Book chapter, 2019)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1= Achar, C., So, J., Agrawal, N., &amp; Duhachek, A. (2016). What we feel and why we buy: the influence of emotions on consumer decision-making. Current opinion in psychology, 10, 166-170. https://doi.org/10.1016/j.copsyc.2016.01.009

Bigné, J. E., &amp; Andreu, L. (2004). Emotions in segmentation: An empirical study. Annals of Tourism Research, 31(3), 682-696. https://doi.org/10.1016/j.annals.2003.12.018

Chamberlain, L., &amp; Broderick, A. J. (2007). The application of physiological observation methods to emotion research. Qualitative Market Research: An International Journal, 10(2), 199-216. https://doi.org/10.1108/13522750710740853

Ekman, P., Dalgleish, T., &amp; Power, M. (1999). Basic emotions. ''San Francisco, USA'', ''1''.

Faullant, R., Matzler, K., &amp; Mooradian, T. A. (2011). Personality, basic emotions, and satisfaction: Primary emotions in the mountaineering experience. Tourism management, 32(6), 1423-1430. https://doi.org/10.1016/j.tourman.2011.01.004

Harmon-Jones, E., Harmon-Jones, C., &amp; Summerell, E. (2017). On the importance of both dimensional and discrete models of emotion. Behavioral sciences, 7(4), 66. https://doi.org/10.3390/bs7040066

James, W. (2002). The James-Lange theory of emotion. Visceral sensory neuroscience: interoception, 9.

Kranzbühler, A. M., Zerres, A., Kleijnen, M. H., &amp; Verlegh, P. W. (2020). Beyond valence: A meta-analysis of discrete emotions in firm-customer encounters. Journal of the Academy of Marketing Science, 48(3), 478-498. https://doi.org/10.1007/s11747-019-00707-0

Lerner, J. S., &amp; Keltner, D. (2001). Fear, anger, and risk. Journal of personality and social psychology, 81(1), 146. https://doi.org/10.1037/0022-3514.81.1.146

Matsumoto, D., &amp; Hwang, H. S. (2012). Culture and emotion: The integration of biological and cultural contributions. Journal of Cross-Cultural Psychology, 43(1), 91-118. https://doi.org/10.1177/0022022111420147

Ningjian, L. (2025). James-Lange Theory of Emotion. In The ECPH Encyclopedia of Psychology (pp. 774-775). Singapore: Springer Nature Singapore. https://doi.org/10.1007/978-981-97-7874-4_1215

PS, S., &amp; Mahalakshmi, G. (2017). Emotion models: a review. International Journal of Control Theory and Applications, 10(8), 651-657.

Stein, N. L., &amp; Trabasso, T. (1992). The organisation of emotional experience: Creating links among emotion, thinking, language, and intentional action. ''Cognition &amp; Emotion'', ''6''(3-4), 225-244. https://doi.org/10.1080/02699939208411070

Xie, C., Bagozzi, R. P., &amp; Grønhaug, K. (2015). The role of moral emotions and individual differences in consumer responses to corporate green and non-green actions. Journal of the academy of Marketing Science, 43(3), 333-356. https://doi.org/10.1007/s11747-014-0394-5

Zeelenberg, M., &amp; Pieters, R. (2004). Beyond valence in customer dissatisfaction: A review and new findings on behavioral responses to regret and disappointment in failed services. Journal of business Research, 57(4), 445-455. https://doi.org/10.1016/S0148-2963(02)00278-3


}}
* [[Motivation and emotion/Assessment/Topic|Topic development guidelines]]
** [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
{{tip|Suggestions for this section:
* Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
** Use &quot;Edit source&quot;
** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
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** doi as a URL which is a working hyperlink (i.e., clickable)
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
* [https://www.ted.com/talks/sam_usher_neuromarketing_knowing_why_you_buy Neuromarketing: Knowing Why You Buy | Sam Usher | TED talk] (TED)
* [https://www.apa.org/news/podcasts/speaking-of-psychology/shopping-behavior Tightwads and spendthrifts: How emotions drive our shopping behaviour | Podcast] (Speaking of Psychology, American Psychological Association)
* [https://www.ted.com/talks/lisa_feldman_barrett_you_aren_t_at_the_mercy_of_your_emotions_your_brain_creates_them You aren't at the mercy of your emotions -- your brain creates them | Lisa Feldman Barrett | TED talk] (TED)</text>
      <sha1>fvddre55o712qbrhfke1twhasbmlywv</sha1>
    </revision>
  </page>
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    <title>Motivation and emotion/Book/2026/Attachment styles and relatedness motivation</title>
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      <text bytes="13839" sha1="pmq6uq6ebfxf1tegaid9lao39bbaqyr" xml:space="preserve">{{title|Attachment styles and relatedness motivation:&lt;br&gt; How do attachment styles affect the need for relatedness?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
;Scenario

Harry and Caitlin have both recently started university and are adjusting to their new environment. 
[[File:People and stands at the Museum Reserve, Orientation Week MS-5342 012 002.jpg|thumb|'''Figure 1'''. Insert caption that illustrates scenario]]

Harry throws himself into orientation week, joining clubs and making an effort to sit next to people he recognises. However, when plans are made without him or his messages go unanswered, he wonders whether people actually like him. After hanging out with some new friends, he replays conversations and wonders what he could have done differently

Caitlin chats easily with people at events and joins a few clubs, but she often turns down invitations to catch up outside of university. When conversations turn more personal or people begin trying to get to know her better, she makes an excuse to leave early and may decide not to attend next time. Although she enjoys being included, she finds keeping some distance more comfortable. 

{{RoundBoxBottom}}

* Explain relatedness as a basic psychological need within self-determinantion theory and briefly explain its importance for psychological functioning, belonging, motivation, and wellbeing
** e.g. feeling connected, accepted, and that we belong is associated with better ... '''(source)'''
* Briefly introduce attachment theory (remebering to keep it shallow as we delve deeper into definiton and explanations in the section) and how people may differ in how the need for relatedness is experienced and expressed
* central issue of the chapter;
{{RoundBoxTop|theme=2}}
I don't know if i'm thinking to deeply into the chapter question, but the way i want to answer it consists of two parts three parts
# Do attachment styles effect the need for relatedness
# Do different attachment styles affect the strength of the need itself or the way it's regulated and expressed
hard to argue/look into how attachment styles affect the need for relatedness with establishing whether or not there is an effect in the first place.
{{RoundBoxBottom}}
* outline the aim of the chapter: to draw on psychological theory and research to examine how secure, anxious, avoidant and fearful-avoidant attachment styles may influence relatedness motivation, and  the reverse (whether experiences of relatedness may also contribute to changes in attachment over time). '''''probably not necessary but sounds interesting to look into'''''

{{RoundBoxTop|theme=1}}

'''Focus questions'''
* How does self-determination theory explain relatedness as a basic psychological need?
* How does attachment theory conceptualise different styles of attachment?
* How are attachment styles understood in adulthood?
* How do different attachment styles influence relatedness motivation?
* How might experiences of relatedness shape attachment over time?

{{RoundBoxBottom}}

==Headings==
Use this heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External links

==Self-determination theory==

===Autonomy===
===Competence===
===Relatedness===


==Attachment theory and attachment styles==
* Attachment thoery was first developed by British Psychoanalyst John Bowlby, who sought to understand the anxiety and distress infants experience when separated from their primary caregivers
* Bowlby observed that infants use attachment behaviours (e.g. crying, clinging, following) to prevent separation from, or re-establish proximity to, their caregivers. (Bowlby, 1958)
* He argued that these attachment behaviours were evolutionary and adaptive functions; infants who were able to maintain proximity to an attachment figure (someone who provides support, protection, and care) would be more likely to survive to reproductive age, making proximity-seeking a survival mechanism; He coined this the attachment behavioural system, an innate psychobiological mechanism that motivates individuals to seek and maintain closeness to a primary caregiver or attachment figure for safety and protection (Bowlby, 1982)
* Although Bowlby considered the attachment system to be universal, he recognised that infants could differ in how attachment was experienced and expressed; He theorised that early attachment experiences with caregivers contribute to an internal working model (a mental representations of expectations of whether attachment figures are available and responsive, and whether the self is worthy of care and support); An infants internal working model is thought to guide how they respond to situations that activate their attachment system (i.e. distress, separation), expectations on whether support will be available influence whether the infant seeks proximity and reassurance, intensifies attachment behaviours, or withdraws and suppresses attachment needs (i.e. the quality/type of attachment that an infant develops is largely determined by the caregiver’s response to the infant when the infant’s attachment system is ‘activated’) (Bretherton &amp; Munholland, 2008; Collins &amp; Allard, 2004; Sherman et al., 2015)
* While Bowlby provided a theoretical understanding, Mary Ainsworth extended his work by examining how differences in attachment were expressed in infant's behaviour.
* Mary Ainsworth was an American-Canadian developmental psychologist who developed the [https://en.wikipedia.org/wiki/Strange_situation Strange Situation procedure], which involved observing infants during brief separations from and reunions with their caregivers
* During the procedure, attention is paid to four aspects of the infants behaviour (Ainsworth et al., 1978):
** The amount of exploration the infant engaged in (e.g. playing with new toys) throughout.
** The infant's reactions to the departure of their caregiver.
** The infant's response to being alone with a stranger.
** The infant's behaviour when reunited with their caregiver.
* Based on these observations, the infants were categorised into three groups that would become foundation of attachment styes.

{{RoundBoxTop|theme=4}}
considering breaking this part into subsections (i.e. attachment behavioural systems, internal working model, Mary Ainsworth and strange situation procedure). However would that be too many (would it improve readability/clarity?) Additionally, this overview of theory may be too deep (i.e. is all this info necessary, am I straying from the main focus)
{{RoundBoxBottom}}

===Secure attachment===
===Anxious-ambivalent attachment===
===Anxious-avoidant attachment===
===Disorganised-disoriented attachment===
===Attachment in adulthood===
{{RoundBoxTop|theme=4}}
should this be a level 3 heading?
{{RoundBoxBottom}}

==Attachment styles and the need for relatedness==


==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[wikipedia:Anxious-preoccupied_attachment|Anxious attachment]] (Wikipedia)
* [[wikipedia:Attachment_theory|Attachment theory]] (Wikipedia)
* [[Self-determination theory]] (Wikiversity)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Ainsworth, M. D. S., Blehar, M. C., Waters, E., &amp; Wall, S. (1978). ''Patterns of attachment: A psychological study of the strange situation''. Lawrence Erlbaum. 

Bowlby, J. (1958). The nature of the child's tie to his mother. ''Int J Psychoanal'', 39(5), 350-373. 

Bowlby, J. (1982). Attachment and loss: Retrospect and prospect. ''American Journal of Orthopsychiatry'', 52(4), 664-678. https://doi.org/10.1111/j.1939-0025.1982.tb01456.x 

Bretherton, I., &amp; Munholland, K. A. (2008). Internal working models in attachment relationships: Elaborating a central construct in attachment theory. In ''Handbook of attachment: Theory, research, and clinical applications'', 2nd ed. (pp. 102-127). The Guilford Press.

Collins, N. L., &amp; Allard, L. M. (2004). ''Cognitive Representations of Attachment: The Content and Function of Working Models''. Blackwell Publishing.

Sherman, L. J., Rice, K., &amp; Cassidy, J. (2015). Infant capacities related to building internal working models of attachment figures: A theoretical and empirical review. ''Developmental Review'', 37, 109-141. https://doi.org/https://doi.org/10.1016/j.dr.2015.06.001 


}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relationships/Attachment]]
[[Category:Motivation and emotion/Book/Needs/Psychological/Relatedness]]</text>
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__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
;Scenario

Harry and Caitlin have both recently started university and are adjusting to their new environment. 
[[File:People and stands at the Museum Reserve, Orientation Week MS-5342 012 002.jpg|thumb|'''Figure 1'''. Insert caption that illustrates scenario]]

Harry throws himself into orientation week, joining clubs and making an effort to sit next to people he recognises. However, when plans are made without him or his messages go unanswered, he wonders whether people actually like him. After hanging out with some new friends, he replays conversations and wonders what he could have done differently

Caitlin chats easily with people at events and joins a few clubs, but she often turns down invitations to catch up outside of university. When conversations turn more personal or people begin trying to get to know her better, she makes an excuse to leave early and may decide not to attend next time. Although she enjoys being included, she finds keeping some distance more comfortable. 

{{RoundBoxBottom}}

* Explain relatedness as a basic psychological need within self-determinantion theory and briefly explain its importance for psychological functioning, belonging, motivation, and wellbeing
** e.g. feeling connected, accepted, and that we belong is associated with better ... '''(source)'''
* Briefly introduce attachment theory (remebering to keep it shallow as we delve deeper into definiton and explanations in the section) and how people may differ in how the need for relatedness is experienced and expressed
* central issue of the chapter;
{{RoundBoxTop|theme=2}}
I don't know if i'm thinking to deeply into the chapter question, but the way i want to answer it consists of two parts three parts
# Do attachment styles effect the need for relatedness
# Do different attachment styles affect the strength of the need itself or the way it's regulated and expressed
hard to argue/look into how attachment styles affect the need for relatedness with establishing whether or not there is an effect in the first place.
{{RoundBoxBottom}}
* outline the aim of the chapter: to draw on psychological theory and research to examine how secure, anxious, avoidant and fearful-avoidant attachment styles may influence relatedness motivation, and  the reverse (whether experiences of relatedness may also contribute to changes in attachment over time). '''''probably not necessary but sounds interesting to look into'''''

{{RoundBoxTop|theme=1}}

'''Focus questions'''
* How does self-determination theory explain relatedness as a basic psychological need?
* How does attachment theory conceptualise different styles of attachment?
* How are attachment styles understood in adulthood?
* How do different attachment styles influence relatedness motivation?
* How might experiences of relatedness shape attachment over time?

{{RoundBoxBottom}}

==Headings==
Use this heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings, but:
** avoid having only one sub-heading
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* See also
* References
* External links

==Self-determination theory==

===Autonomy===
===Competence===
===Relatedness===


==Attachment theory and attachment styles==
* Attachment thoery was first developed by British Psychoanalyst John Bowlby, who sought to understand the anxiety and distress infants experience when separated from their primary caregivers
* Bowlby observed that infants use attachment behaviours (e.g. crying, clinging, following) to prevent separation from, or re-establish proximity to, their caregivers. (Bowlby, 1958)
* He argued that these attachment behaviours were evolutionary and adaptive functions; infants who were able to maintain proximity to an attachment figure (someone who provides support, protection, and care) would be more likely to survive to reproductive age, making proximity-seeking a survival mechanism; He coined this the attachment behavioural system, an innate psychobiological mechanism that motivates individuals to seek and maintain closeness to a primary caregiver or attachment figure for safety and protection (Bowlby, 1982)
* Although Bowlby considered the attachment system to be universal, he recognised that infants could differ in how attachment was experienced and expressed; He theorised that early attachment experiences with caregivers contribute to an internal working model (a mental representations of expectations of whether attachment figures are available and responsive, and whether the self is worthy of care and support)
* ;An infants internal working model is thought to guide how they respond to situations that activate their attachment system (i.e. distress, separation), 
* Beginning at 6 months, infants start to anticipate specific caregivers responses to their distress and shape their own behaviour accordingly (Sroufe, 1988); expectations on whether support will be available influence whether the infant seeks proximity and reassurance, intensifies attachment behaviours, or withdraws and suppresses attachment needs (i.e. the quality/type of attachment that an infant develops is largely determined by the caregiver’s response to the infant when the infant’s attachment system is ‘activated’) (Bretherton &amp; Munholland, 2008; Collins &amp; Allard, 2004; Sherman et al., 2015)
* While Bowlby provided a theoretical understanding, Mary Ainsworth extended his work by examining how differences in attachment were expressed in infant's behaviour.
* Mary Ainsworth was an American-Canadian developmental psychologist who developed the [https://en.wikipedia.org/wiki/Strange_situation Strange Situation procedure], which involved observing infants during brief separations from and reunions with their caregivers
* During the procedure, attention is paid to four aspects of the infants behaviour (Ainsworth et al., 1978):
** The amount of exploration the infant engaged in (e.g. playing with new toys) throughout.
** The infant's reactions to the departure of their caregiver.
** The infant's response to being alone with a stranger.
** The infant's behaviour when reunited with their caregiver.
* Based on these observations, the infants were categorised into three groups that would become foundation of attachment styes.

{{RoundBoxTop|theme=4}}
considering breaking this part into subsections (i.e. attachment behavioural systems, internal working model, Mary Ainsworth and strange situation procedure). However would that be too many (would it improve readability/clarity?) Additionally, this overview of theory may be too deep (i.e. is all this info necessary, am I straying from the main focus)
{{RoundBoxBottom}}

===Secure attachment===

* Typical behaviour '''(Source)'''
** uses the caregiver as a secure base, exploring independently while checking back or returning to caregiver when needed
** seeks caregiver when attachment behavioural system is activated (i.e. when distressed, hurt, tired, etc.)
** usually accepts comfort and settles once reassurance is provided
* Strange Situation (Group B) (Perasso, 2023) '''(Source)'''
** Balances exploration of the environment with maintaing proximity to caregiver
** moderate distress by separation from caregiver, on reunion seeks or welcomes contact from caregiver and accepts comfort and reassurance
** typically return to exploration once settled
* Developmental Factors/Caregiver response '''(Source)'''
** associated with caregiving that is usually available and responds appropriately (generally sensitive, responsive, and consistent) when the infant signals distress or a need for support
** Internal Working Model '''(Source)'''
*** Repeated experiences of having needs responded to positively may lead the infant to expect that the caregiver will be available and responsive when support is required.
*** The infant can therefore express distress and seek proximity without needing to excessively intensify or suppress attachment behaviour.

===Anxious-ambivalent attachment===

* Typical behaviour '''(Source)'''
** May remain highly attentive to the caregiver rather than exploring freely.
** Can become easily distressed by separation or uncertainty about the caregiver's availability, seeking frequent proximity, attention, or reassurance.
** Can have difficulty settling even when comfort is offered.
** May appear clingy while also showing frustration, anger, or resistance toward the caregiver.
* Strange Situation (Group C) (Perasso, 2023) '''(Source)'''
** Typically show absent/low interest in exploring environment throughout procedure,
** intensely distressed and visibly upset once separated from caregiver
** Extreme reactions of fear and distress toward the stranger
** During reunion are difficult to calm down even when picked up and reassured by caregiver, may initially seek comfort then resist or reject caregivers effort to comfort them
* Developmental Factors/Caregiver response '''(Source)'''
** Associated with caregiving that may be inconsistent or unpredictable.
** The caregiver may sometimes respond to distress but may be unavailable or less responsive at other times.
** Internal working model '''(Source)'''
*** Availability of caregiver is experienced as uncertain or unpredictable. 
*** Expectation that support cannot be consistently available develops, causing infant to become particularly attentive to the caregiver’s availability and intensify attachment signals in an attempt to obtain a response '''(maximising or hyperactivating attachment strategy)'''

===Anxious-avoidant attachment===

* Typical behaviour '''(Source)'''
** May appear unusually independent and spend more time focused on objects, activities, or exploration than on the caregiver
** May show relatively little outward distress when hurt, upset, or separated despite still experiencing physiological or emotional distress
** Is less likely to openly seek comfort or reassurance from the caregiver, may turn away, ignore, or minimise contact even when distressed
** Tends to suppress or minimise visible attachment signals
* Strange Situation (Group A) (Perasso, 2023) '''(source)'''
** often remain focuses on exploration rather than proximity to caregiver
** show mininmal or no distress by the separation from caregiver
** May appear relatively unaffected by the presence of the stranger compared with other attachment classifications.
** show limited interest or excitement upon reunion with caregiver, often ignoring or avoiding interactions
* Developmental Factors/Caregiver response '''(Source)'''
** Associated with caregiving that may be relatively unresponsive, rejecting, or uncomfortable with expressions of distress and bids for comfort.
*** Internal working model '''(Source)'''
**** infant may have developed an expectation that attempts to obtain comfort or support are unlikely to be met with a responsive reaction. 
**** As a result, attachment signals may be minimised or suppressed ('''deactivating or minimising attachment strategy)'''.

===Disorganised-disoriented attachment===

* Disorganised-disoriented attachment was introduced later by Main and Solomon (1986).
* The classification emerged from observations of infant behaviours that did not fit comfortably within Ainsworth’s original secure, resistant, and avoidant classifications.
* Typical behaviour '''(Source)'''
** Behaviour toward the caregiver may be inconsistent or difficult to predict, can show unusual combinations of proximity-seeking and withdrawal.
** May have difficulty developing a consistent strategy for obtaining comfort when distressed.
** Behaviour can become particularly confused or disorganised during situations involving fear, stress, or uncertainty.
* Strange Situation (Duschinsky, 2018)
** Display  contradictory, confused, fearful, or disoriented behaviours, particularly during reunion with the caregiver, e.g. may aproach caregiver and then withdraw, freeze, show incomplete or misdirected movements, unusual postures, or signs of apprehension toward the caregiver.
* Developmental Factors/Caregiver response '''(Source)'''
** Has been associated with caregiving that is frightening, frightened, highly disrupted, or unpredictable.
** It is also found at higher rates in some contexts involving significant adversity '''(Source)'''
*** Internal working model '''(Source)'''
**** more difficult to explain through one coherent internal working model because the infant does not appear to have a consistent strategy for obtaining safety.
**** One proposed explanation is that the caregiver may simultaneously represent a potential source of safety and a source of fear or uncertainty, creating conflict between approaching the caregiver for protection and moving away from them.

===Attachment in adulthood===
{{RoundBoxTop|theme=4}}
should this be a level 3 heading?
{{RoundBoxBottom}}

==Attachment styles and the need for relatedness==


==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[wikipedia:Anxious-preoccupied_attachment|Anxious attachment]] (Wikipedia)
* [[wikipedia:Attachment_theory|Attachment theory]] (Wikipedia)
* [[Self-determination theory]] (Wikiversity)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Ainsworth, M. D. S., Blehar, M. C., Waters, E., &amp; Wall, S. (1978). ''Patterns of attachment: A psychological study of the strange situation''. Lawrence Erlbaum. 

Bowlby, J. (1958). The nature of the child's tie to his mother. ''Int J Psychoanal'', 39(5), 350-373. 

Bowlby, J. (1982). Attachment and loss: Retrospect and prospect. ''American Journal of Orthopsychiatry'', 52(4), 664-678. https://doi.org/10.1111/j.1939-0025.1982.tb01456.x 

Bretherton, I., &amp; Munholland, K. A. (2008). Internal working models in attachment relationships: Elaborating a central construct in attachment theory. In ''Handbook of attachment: Theory, research, and clinical applications'', 2nd ed. (pp. 102-127). The Guilford Press.

Collins, N. L., &amp; Allard, L. M. (2004). ''Cognitive Representations of Attachment: The Content and Function of Working Models''. Blackwell Publishing.

Sherman, L. J., Rice, K., &amp; Cassidy, J. (2015). Infant capacities related to building internal working models of attachment figures: A theoretical and empirical review. ''Developmental Review'', 37, 109-141. https://doi.org/https://doi.org/10.1016/j.dr.2015.06.001 


}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relationships/Attachment]]
[[Category:Motivation and emotion/Book/Needs/Psychological/Relatedness]]</text>
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        <id>3106307</id>
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__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
;Scenario

Harry and Caitlin have both recently started university and are adjusting to their new environment. 
[[File:People and stands at the Museum Reserve, Orientation Week MS-5342 012 002.jpg|thumb|'''Figure 1'''. Insert caption that illustrates scenario]]

Harry throws himself into orientation week, joining clubs and making an effort to sit next to people he recognises. However, when plans are made without him or his messages go unanswered, he wonders whether people actually like him. After hanging out with some new friends, he replays conversations and wonders what he could have done differently

Caitlin chats easily with people at events and joins a few clubs, but she often turns down invitations to catch up outside of university. When conversations turn more personal or people begin trying to get to know her better, she makes an excuse to leave early and may decide not to attend next time. Although she enjoys being included, she finds keeping some distance more comfortable. 

{{RoundBoxBottom}}

* Explain relatedness as a basic psychological need within self-determinantion theory and briefly explain its importance for psychological functioning, belonging, motivation, and wellbeing
** e.g. feeling connected, accepted, and that we belong is associated with better ... '''(source)'''
* Briefly introduce attachment theory (remebering to keep it shallow as we delve deeper into definiton and explanations in the section) and how people may differ in how the need for relatedness is experienced and expressed
* central issue of the chapter;
{{RoundBoxTop|theme=2}}
I don't know if i'm thinking to deeply into the chapter question, but the way i want to answer it consists of two parts three parts
# Do attachment styles effect the need for relatedness
# Do different attachment styles affect the strength of the need itself or the way it's regulated and expressed
hard to argue/look into how attachment styles affect the need for relatedness with establishing whether or not there is an effect in the first place.
{{RoundBoxBottom}}
* outline the aim of the chapter: to draw on psychological theory and research to examine how secure, anxious, avoidant and fearful-avoidant attachment styles may influence relatedness motivation, and  the reverse (whether experiences of relatedness may also contribute to changes in attachment over time). '''''probably not necessary but sounds interesting to look into'''''

{{RoundBoxTop|theme=1}}

'''Focus questions'''
* How does self-determination theory explain relatedness as a basic psychological need?
* How does attachment theory conceptualise different styles of attachment?
* How are attachment styles understood in adulthood?
* How do different attachment styles influence relatedness motivation?
* How might experiences of relatedness shape attachment over time?

{{RoundBoxBottom}}

==Self-determination theory==

===Autonomy===
===Competence===
===Relatedness===


==Attachment theory and attachment styles==
* Attachment thoery was first developed by British Psychoanalyst John Bowlby, who sought to understand the anxiety and distress infants experience when separated from their primary caregivers
* Bowlby observed that infants use attachment behaviours (e.g. crying, clinging, following) to prevent separation from, or re-establish proximity to, their caregivers. (Bowlby, 1958)
* He argued that these attachment behaviours were evolutionary and adaptive functions; infants who were able to maintain proximity to an attachment figure (someone who provides support, protection, and care) would be more likely to survive to reproductive age, making proximity-seeking a survival mechanism; He coined this the attachment behavioural system, an innate psychobiological mechanism that motivates individuals to seek and maintain closeness to a primary caregiver or attachment figure for safety and protection (Bowlby, 1982)
* Although Bowlby considered the attachment system to be universal, he recognised that infants could differ in how attachment was experienced and expressed; He theorised that early attachment experiences with caregivers contribute to an internal working model (a mental representations of expectations of whether attachment figures are available and responsive, and whether the self is worthy of care and support)
* ;An infants internal working model is thought to guide how they respond to situations that activate their attachment system (i.e. distress, separation), 
* Beginning at 6 months, infants start to anticipate specific caregivers responses to their distress and shape their own behaviour accordingly (Sroufe, 1988); expectations on whether support will be available influence whether the infant seeks proximity and reassurance, intensifies attachment behaviours, or withdraws and suppresses attachment needs (i.e. the quality/type of attachment that an infant develops is largely determined by the caregiver’s response to the infant when the infant’s attachment system is ‘activated’) (Bretherton &amp; Munholland, 2008; Collins &amp; Allard, 2004; Sherman et al., 2015)
* While Bowlby provided a theoretical understanding, Mary Ainsworth extended his work by examining how differences in attachment were expressed in infant's behaviour.
* Mary Ainsworth was an American-Canadian developmental psychologist who developed the [https://en.wikipedia.org/wiki/Strange_situation Strange Situation procedure], which involved observing infants during brief separations from and reunions with their caregivers
* During the procedure, attention is paid to four aspects of the infants behaviour (Ainsworth et al., 1978):
** The amount of exploration the infant engaged in (e.g. playing with new toys) throughout.
** The infant's reactions to the departure of their caregiver.
** The infant's response to being alone with a stranger.
** The infant's behaviour when reunited with their caregiver.
* Based on these observations, the infants were categorised into three groups that would become foundation of attachment styes.

{{RoundBoxTop|theme=4}}
considering breaking this part into subsections (i.e. attachment behavioural systems, internal working model, Mary Ainsworth and strange situation procedure). However would that be too many (would it improve readability/clarity?) Additionally, this overview of theory may be too deep (i.e. is all this info necessary, am I straying from the main focus)
{{RoundBoxBottom}}

===Secure attachment===

* Typical behaviour '''(Source)'''
** uses the caregiver as a secure base, exploring independently while checking back or returning to caregiver when needed
** seeks caregiver when attachment behavioural system is activated (i.e. when distressed, hurt, tired, etc.)
** usually accepts comfort and settles once reassurance is provided
* Strange Situation (Group B) (Perasso, 2023) '''(Source)'''
** Balances exploration of the environment with maintaing proximity to caregiver
** moderate distress by separation from caregiver, on reunion seeks or welcomes contact from caregiver and accepts comfort and reassurance
** typically return to exploration once settled
* Developmental Factors/Caregiver response '''(Source)'''
** associated with caregiving that is usually available and responds appropriately (generally sensitive, responsive, and consistent) when the infant signals distress or a need for support
** Internal Working Model '''(Source)'''
*** Repeated experiences of having needs responded to positively may lead the infant to expect that the caregiver will be available and responsive when support is required.
*** The infant can therefore express distress and seek proximity without needing to excessively intensify or suppress attachment behaviour.

===Anxious-ambivalent attachment===

* Typical behaviour '''(Source)'''
** May remain highly attentive to the caregiver rather than exploring freely.
** Can become easily distressed by separation or uncertainty about the caregiver's availability, seeking frequent proximity, attention, or reassurance.
** Can have difficulty settling even when comfort is offered.
** May appear clingy while also showing frustration, anger, or resistance toward the caregiver.
* Strange Situation (Group C) (Perasso, 2023) '''(Source)'''
** Typically show absent/low interest in exploring environment throughout procedure,
** intensely distressed and visibly upset once separated from caregiver
** Extreme reactions of fear and distress toward the stranger
** During reunion are difficult to calm down even when picked up and reassured by caregiver, may initially seek comfort then resist or reject caregivers effort to comfort them
* Developmental Factors/Caregiver response '''(Source)'''
** Associated with caregiving that may be inconsistent or unpredictable.
** The caregiver may sometimes respond to distress but may be unavailable or less responsive at other times.
** Internal working model '''(Source)'''
*** Availability of caregiver is experienced as uncertain or unpredictable. 
*** Expectation that support cannot be consistently available develops, causing infant to become particularly attentive to the caregiver’s availability and intensify attachment signals in an attempt to obtain a response '''(maximising or hyperactivating attachment strategy)'''

===Anxious-avoidant attachment===

* Typical behaviour '''(Source)'''
** May appear unusually independent and spend more time focused on objects, activities, or exploration than on the caregiver
** May show relatively little outward distress when hurt, upset, or separated despite still experiencing physiological or emotional distress
** Is less likely to openly seek comfort or reassurance from the caregiver, may turn away, ignore, or minimise contact even when distressed
** Tends to suppress or minimise visible attachment signals
* Strange Situation (Group A) (Perasso, 2023) '''(source)'''
** often remain focuses on exploration rather than proximity to caregiver
** show mininmal or no distress by the separation from caregiver
** May appear relatively unaffected by the presence of the stranger compared with other attachment classifications.
** show limited interest or excitement upon reunion with caregiver, often ignoring or avoiding interactions
* Developmental Factors/Caregiver response '''(Source)'''
** Associated with caregiving that may be relatively unresponsive, rejecting, or uncomfortable with expressions of distress and bids for comfort.
*** Internal working model '''(Source)'''
**** infant may have developed an expectation that attempts to obtain comfort or support are unlikely to be met with a responsive reaction. 
**** As a result, attachment signals may be minimised or suppressed ('''deactivating or minimising attachment strategy)'''.

===Disorganised-disoriented attachment===

* Disorganised-disoriented attachment was introduced later by Main and Solomon (1986).
* The classification emerged from observations of infant behaviours that did not fit comfortably within Ainsworth’s original secure, resistant, and avoidant classifications.
* Typical behaviour '''(Source)'''
** Behaviour toward the caregiver may be inconsistent or difficult to predict, can show unusual combinations of proximity-seeking and withdrawal.
** May have difficulty developing a consistent strategy for obtaining comfort when distressed.
** Behaviour can become particularly confused or disorganised during situations involving fear, stress, or uncertainty.
* Strange Situation (Duschinsky, 2018)
** Display  contradictory, confused, fearful, or disoriented behaviours, particularly during reunion with the caregiver, e.g. may aproach caregiver and then withdraw, freeze, show incomplete or misdirected movements, unusual postures, or signs of apprehension toward the caregiver.
* Developmental Factors/Caregiver response '''(Source)'''
** Has been associated with caregiving that is frightening, frightened, highly disrupted, or unpredictable.
** It is also found at higher rates in some contexts involving significant adversity '''(Source)'''
*** Internal working model '''(Source)'''
**** more difficult to explain through one coherent internal working model because the infant does not appear to have a consistent strategy for obtaining safety.
**** One proposed explanation is that the caregiver may simultaneously represent a potential source of safety and a source of fear or uncertainty, creating conflict between approaching the caregiver for protection and moving away from them.

{{RoundBoxTop|theme=4}}
potential information to be added for book chapter
* more information in development of each attachment style outside of caregiver response and IWM
** e.g. temperament and child characteristics, broader family/contextual influences
* criticisms of attachment styles
** whether attachment patterns and Strange Situation behaviour are expressed similarly across cultures
{{RoundBoxBottom}}

===Attachment in adulthood===
{{RoundBoxTop|theme=4}}
should this be a level 3 heading?
{{RoundBoxBottom}}

==Attachment styles and the need for relatedness==


==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[wikipedia:Anxious-preoccupied_attachment|Anxious attachment]] (Wikipedia)
* [[wikipedia:Attachment_theory|Attachment theory]] (Wikipedia)
* [[Self-determination theory]] (Wikiversity)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Ainsworth, M. D. S., Blehar, M. C., Waters, E., &amp; Wall, S. (1978). ''Patterns of attachment: A psychological study of the strange situation''. Lawrence Erlbaum. 

Bowlby, J. (1958). The nature of the child's tie to his mother. ''Int J Psychoanal'', 39(5), 350-373. 

Bowlby, J. (1982). Attachment and loss: Retrospect and prospect. ''American Journal of Orthopsychiatry'', 52(4), 664-678. https://doi.org/10.1111/j.1939-0025.1982.tb01456.x 

Bretherton, I., &amp; Munholland, K. A. (2008). Internal working models in attachment relationships: Elaborating a central construct in attachment theory. In ''Handbook of attachment: Theory, research, and clinical applications'', 2nd ed. (pp. 102-127). The Guilford Press.

Collins, N. L., &amp; Allard, L. M. (2004). ''Cognitive Representations of Attachment: The Content and Function of Working Models''. Blackwell Publishing.

Sherman, L. J., Rice, K., &amp; Cassidy, J. (2015). Infant capacities related to building internal working models of attachment figures: A theoretical and empirical review. ''Developmental Review'', 37, 109-141. https://doi.org/https://doi.org/10.1016/j.dr.2015.06.001 


}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Relationships/Attachment]]
[[Category:Motivation and emotion/Book/Needs/Psychological/Relatedness]]</text>
      <sha1>j9yzakduirp3oz5u0sdg7stu23r2cp1</sha1>
    </revision>
    <revision>
      <id>2830740</id>
      <parentid>2830648</parentid>
      <timestamp>2026-09-03T11:58:21Z</timestamp>
      <contributor>
        <username>HawaSA</username>
        <id>3106307</id>
      </contributor>
      <comment>/* Secure attachment */</comment>
      <origin>2830740</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23203" sha1="jvlaasm9ldm5kmxjoffxz5y5fzkzcq7" xml:space="preserve">{{title|Attachment styles and relatedness motivation:&lt;br&gt; How do attachment styles affect the need for relatedness?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
;Scenario

Harry and Caitlin have both recently started university and are adjusting to their new environment. 
[[File:People and stands at the Museum Reserve, Orientation Week MS-5342 012 002.jpg|thumb|'''Figure 1'''. Insert caption that illustrates scenario]]

Harry throws himself into orientation week, joining clubs and making an effort to sit next to people he recognises. However, when plans are made without him or his messages go unanswered, he wonders whether people actually like him. After hanging out with some new friends, he replays conversations and wonders what he could have done differently

Caitlin chats easily with people at events and joins a few clubs, but she often turns down invitations to catch up outside of university. When conversations turn more personal or people begin trying to get to know her better, she makes an excuse to leave early and may decide not to attend next time. Although she enjoys being included, she finds keeping some distance more comfortable. 

{{RoundBoxBottom}}

* Explain relatedness as a basic psychological need within self-determinantion theory and briefly explain its importance for psychological functioning, belonging, motivation, and wellbeing
** e.g. feeling connected, accepted, and that we belong is associated with better ... '''(source)'''
* Briefly introduce attachment theory (remebering to keep it shallow as we delve deeper into definiton and explanations in the section) and how people may differ in how the need for relatedness is experienced and expressed
* central issue of the chapter;
{{RoundBoxTop|theme=2}}
I don't know if i'm thinking to deeply into the chapter question, but the way i want to answer it consists of two parts three parts
# Do attachment styles effect the need for relatedness
# Do different attachment styles affect the strength of the need itself or the way it's regulated and expressed
hard to argue/look into how attachment styles affect the need for relatedness with establishing whether or not there is an effect in the first place.
{{RoundBoxBottom}}
* outline the aim of the chapter: to draw on psychological theory and research to examine how secure, anxious, avoidant and fearful-avoidant attachment styles may influence relatedness motivation, and  the reverse (whether experiences of relatedness may also contribute to changes in attachment over time). '''''probably not necessary but sounds interesting to look into'''''

{{RoundBoxTop|theme=1}}

'''Focus questions'''
* How does self-determination theory explain relatedness as a basic psychological need?
* How does attachment theory conceptualise different styles of attachment?
* How are attachment styles understood in adulthood?
* How do different attachment styles influence relatedness motivation?
* How might experiences of relatedness shape attachment over time?

{{RoundBoxBottom}}

==Self-determination theory==

* Self-determination theory (SDT) is an approach to human motivation that seeks to explain what motivates people to act and the conditions that support or undermine motivation.
* Developed by Ryan and Deci (2002), SDT aims to explain human motivation by integrating theories that emphasised natural human growth (humanistic, psychoanalytic, and developmental theories) with those that focused on how behaviour is shaped by environmental and cognitive factors (behavioural and cognitive perspectives)
* SDT assumes that individuals have an inherent tendency towards psychological growth, mastery, and connection with others, that the desire for growth drives behaviour (Ryan &amp; Deci, 2002)
* However this growth is not automatic, SDT proposes that for healthy psychological growth and wellbeing depends on the satisfaction of three basic psychological needs; autonomy, competence, and relatedness. 
* Additionally, a central feature of SDT is that motivation differs not only in amount, but also in quality
* '''Amotivation''': refers to a lack of intention or motivation to engage in a behaviour, potentially due to (Ryan &amp; Deci, 2020)
* '''Intrinsic Motivation''': refers to engaging in a behaviour or activity because it is inherently interesting, enjoyable, or satisfying (Deci &amp; Ryan, 2000; Ryan &amp; Deci, 2020)
** SDT posits that satisfaction of the basic psychological needs for autonomy, competence, and relatedness helps support and maintain intrinsic motivation, contributing to greater engagement, persistence, and satisfaction.
* '''Extrinsic Motivation''': refers to “behaviours done for reasons other that their inherent satisfaction” (Ryan &amp; Deci, 2020)
** SDT doesn’t treat all extrinsic motivation as equal. Instead it breaks it into 4 subtypes according to how autonomous (i.e. doing an activity of your own free will because it feels personally important, valuable, or enjoyable) the motivation is(Ryan &amp; Deci, 2020)
*** External regulation: behaviours driven by externally imposed rewards and punishments
*** Introjected regulation: extrinsic motivation that has been partially internalised, i.e. behaviour regulated by internal rewards of self-esteem (need for success) and avoidance of anxiety, shame, fear of failure
*** Identified regulation: behaviours driven by personal importance/value of activity
*** Integrated regulation: most autonomous/intrinsic form of extrinsic motivation, behaviour driven by not only recognition of the value of the behaviour, but finds it congruent with core interests and values

* Motivation within SDT can therefore be understood along a continuum of self-determination, based on the extent to which behaviour is experienced as autonomous
* Within SDT, the three basic psychological needs help determine the quality of a person’s motivation.
* When these needs are supported, people are more likely to experience intrinsic motivation and more autonomous forms of extrinsic motivation. When the needs are frustrated, motivation may become more controlled, less persistent, or decline altogether. (Deci &amp; Ryan, 2000; Ryan &amp; Deci, 2020)

===Autonomy===
===Competence===
===Relatedness===


==Attachment theory and attachment styles==
* Attachment thoery was first developed by British Psychoanalyst John Bowlby, who sought to understand the anxiety and distress infants experience when separated from their primary caregivers
* Bowlby observed that infants use attachment behaviours (e.g. crying, clinging, following) to prevent separation from, or re-establish proximity to, their caregivers. (Bowlby, 1958)
* He argued that these attachment behaviours were evolutionary and adaptive functions; infants who were able to maintain proximity to an attachment figure (someone who provides support, protection, and care) would be more likely to survive to reproductive age, making proximity-seeking a survival mechanism; He coined this the attachment behavioural system, an innate psychobiological mechanism that motivates individuals to seek and maintain closeness to a primary caregiver or attachment figure for safety and protection (Bowlby, 1982)
* Although Bowlby considered the attachment system to be universal, he recognised that infants could differ in how attachment was experienced and expressed; He theorised that early attachment experiences with caregivers contribute to an internal working model (a mental representations of expectations of whether attachment figures are available and responsive, and whether the self is worthy of care and support)
* ;An infants internal working model is thought to guide how they respond to situations that activate their attachment system (i.e. distress, separation), 
* Beginning at 6 months, infants start to anticipate specific caregivers responses to their distress and shape their own behaviour accordingly (Sroufe, 1988); expectations on whether support will be available influence whether the infant seeks proximity and reassurance, intensifies attachment behaviours, or withdraws and suppresses attachment needs (i.e. the quality/type of attachment that an infant develops is largely determined by the caregiver’s response to the infant when the infant’s attachment system is ‘activated’) (Bretherton &amp; Munholland, 2008; Collins &amp; Allard, 2004; Sherman et al., 2015)
* While Bowlby provided a theoretical understanding, Mary Ainsworth extended his work by examining how differences in attachment were expressed in infant's behaviour.
* Mary Ainsworth was an American-Canadian developmental psychologist who developed the [https://en.wikipedia.org/wiki/Strange_situation Strange Situation procedure], which involved observing infants during brief separations from and reunions with their caregivers
* During the procedure, attention is paid to four aspects of the infants behaviour (Ainsworth et al., 1978):
** The amount of exploration the infant engaged in (e.g. playing with new toys) throughout.
** The infant's reactions to the departure of their caregiver.
** The infant's response to being alone with a stranger.
** The infant's behaviour when reunited with their caregiver.
* Based on these observations, the infants were categorised into three groups that would become foundation of attachment styes.

{{RoundBoxTop|theme=4}}
considering breaking this part into subsections (i.e. attachment behavioural systems, internal working model, Mary Ainsworth and strange situation procedure). However would that be too many (would it improve readability/clarity?) Additionally, this overview of theory may be too deep (i.e. is all this info necessary, am I straying from the main focus)
{{RoundBoxBottom}}

===Secure attachment===

* Typical behaviour (Vaughn &amp; Waters, 1990)
** uses the caregiver as a secure base, exploring independently while checking back or returning to caregiver when needed
** seeks caregiver when attachment behavioural system is activated (i.e. when distressed, hurt, tired, etc.)
** usually accepts comfort and settles once reassurance is provided
* Strange Situation (Group B) (Perasso, 2023; Zeanah et al., 2011)
** Balances exploration of the environment with maintaing proximity to caregiver
** moderate distress by separation from caregiver, on reunion seeks or welcomes contact from caregiver and accepts comfort and reassurance
** typically return to exploration once settled
* Developmental Factors/Caregiver response (Benoit, 2004)
** associated with caregiving that is usually available and responds appropriately (generally sensitive, responsive, and consistent) when the infant signals distress or a need for support
** Internal Working Model (Sherman et al., 2015)
*** Repeated experiences of having needs responded to positively may lead the infant to expect that the caregiver will be available and responsive when support is required.
*** The infant can therefore express distress and seek proximity without needing to excessively intensify or suppress attachment behaviour.

===Anxious-ambivalent attachment===

* Typical behaviour (Benoit, 2004)
** May remain highly attentive to the caregiver rather than exploring freely.
** Can become easily distressed by separation or uncertainty about the caregiver's availability, seeking frequent proximity, attention, or reassurance.
** Can have difficulty settling even when comfort is offered.
** May appear clingy while also showing frustration, anger, or resistance toward the caregiver.
* Strange Situation (Group C) (Perasso, 2023; Zeanah et al., 2011)
** Typically show absent/low interest in exploring environment throughout procedure,
** intensely distressed and visibly upset once separated from caregiver
** Extreme reactions of fear and distress toward the stranger
** During reunion are difficult to calm down even when picked up and reassured by caregiver, may initially seek comfort then resist or reject caregivers effort to comfort them
* Developmental Factors/Caregiver response (Benoit, 2004)
** Associated with caregiving that may be inconsistent or unpredictable.
** The caregiver may sometimes respond to distress but may be unavailable or less responsive at other times.
** Internal working model (Sherman et al., 2015)
*** Availability of caregiver is experienced as uncertain or unpredictable. 
*** Expectation that support cannot be consistently available develops, causing infant to become particularly attentive to the caregiver’s availability and intensify attachment signals in an attempt to obtain a response '''(maximising or hyperactivating attachment strategy)'''

===Anxious-avoidant attachment===

* Typical behaviour (Benoit, 2004)
** May appear unusually independent and spend more time focused on objects, activities, or exploration than on the caregiver
** May show relatively little outward distress when hurt, upset, or separated despite still experiencing physiological or emotional distress
** Is less likely to openly seek comfort or reassurance from the caregiver, may turn away, ignore, or minimise contact even when distressed
** Tends to suppress or minimise visible attachment signals
* Strange Situation (Group A) (Perasso, 2023; Zeanah et al., 2011)
** often remain focuses on exploration rather than proximity to caregiver
** show mininmal or no distress by the separation from caregiver
** May appear relatively unaffected by the presence of the stranger compared with other attachment classifications.
** show limited interest or excitement upon reunion with caregiver, often ignoring or avoiding interactions
* Developmental Factors/Caregiver response (Benoit, 2004)
** Associated with caregiving that may be relatively unresponsive, rejecting, or uncomfortable with expressions of distress and bids for comfort.
*** Internal working model (Sherman et al., 2015)
**** infant may have developed an expectation that attempts to obtain comfort or support are unlikely to be met with a responsive reaction. 
**** As a result, attachment signals may be minimised or suppressed ('''deactivating or minimising attachment strategy)'''.

===Disorganised-disoriented attachment===

* Disorganised-disoriented attachment was introduced later by Main and Solomon (1986).
* The classification emerged from observations of infant behaviours that did not fit comfortably within Ainsworth’s original secure, resistant, and avoidant classifications.
* Typical behaviour (Granqvist et al., 2017)
** Behaviour toward the caregiver may be inconsistent or difficult to predict, can show unusual combinations of proximity-seeking and withdrawal.
** May have difficulty developing a consistent strategy for obtaining comfort when distressed.
** Behaviour can become particularly confused or disorganised during situations involving fear, stress, or uncertainty.
* Strange Situation (Duschinsky, 2018; Zeanah et al., 2011)
** Display  contradictory, confused, fearful, or disoriented behaviours, particularly during reunion with the caregiver, e.g. may aproach caregiver and then withdraw, freeze, show incomplete or misdirected movements, unusual postures, or signs of apprehension toward the caregiver.
* Developmental Factors/Caregiver response (Benoit, 2004)
** Has been associated with caregiving that is frightening, frightened, highly disrupted, or unpredictable.
** It is also found at higher rates in some contexts involving significant adversity (Zeanah et al., 1997)
*** Internal working model (Sherman et al., 2015)
**** more difficult to explain through one coherent internal working model because the infant does not appear to have a consistent strategy for obtaining safety.
**** One proposed explanation is that the caregiver may simultaneously represent a potential source of safety and a source of fear or uncertainty, creating conflict between approaching the caregiver for protection and moving away from them.

{{RoundBoxTop|theme=4}}
potential information to be added for book chapter
* more information in development of each attachment style outside of caregiver response and IWM
** e.g. temperament and child characteristics, broader family/contextual influences
* criticisms of attachment styles
** whether attachment patterns and Strange Situation behaviour are expressed similarly across cultures
{{RoundBoxBottom}}

===Attachment in adulthood===
{{RoundBoxTop|theme=4}}
should this be a level 3 heading?
{{RoundBoxBottom}}

==Attachment styles and the need for relatedness==


==Key points==
For the topic development, for each heading and sub-heading:
* Provide at least three bullet-points, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etc.
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Caption figures (use '''Figure #'''. and explain the relevance of the image to the text)
* Images must be embedded from [[commons:|Wikimedia Commons]]
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples describe concepts in action
* Can be real or fictional; if real, provide citations
* Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Present using [[#Feature boxes|feature boxes]]

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use to tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing Knowing x Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[wikipedia:Anxious-preoccupied_attachment|Anxious attachment]] (Wikipedia)
* [[wikipedia:Attachment_theory|Attachment theory]] (Wikipedia)
* [[Self-determination theory]] (Wikiversity)

{{tip|Suggestions for this section:
* Only select links to major internal resources about the topic
* Include the source in parentheses
}}

==References==
{{Hanging indent|1=

Ainsworth, M. D. S., Blehar, M. C., Waters, E., &amp; Wall, S. (1978). ''Patterns of attachment: A psychological study of the strange situation''. Lawrence Erlbaum. 

Bowlby, J. (1958). The nature of the child's tie to his mother. ''Int J Psychoanal'', 39(5), 350-373. 

Bowlby, J. (1982). Attachment and loss: Retrospect and prospect. ''American Journal of Orthopsychiatry'', 52(4), 664-678. https://doi.org/10.1111/j.1939-0025.1982.tb01456.x 

Bretherton, I., &amp; Munholland, K. A. (2008). Internal working models in attachment relationships: Elaborating a central construct in attachment theory. In ''Handbook of attachment: Theory, research, and clinical applications'', 2nd ed. (pp. 102-127). The Guilford Press.

Collins, N. L., &amp; Allard, L. M. (2004). ''Cognitive Representations of Attachment: The Content and Function of Working Models''. Blackwell Publishing.

Sherman, L. J., Rice, K., &amp; Cassidy, J. (2015). Infant capacities related to building internal working models of attachment figures: A theoretical and empirical review. ''Developmental Review'', 37, 109-141. https://doi.org/https://doi.org/10.1016/j.dr.2015.06.001 


}}

==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Include the source in parentheses after the link
}}

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    <title>Motivation and emotion/Book/2026/Romantic jealousy</title>
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      <comment>wrote up relationship context and uncertainty  GenAi (ChatGPT, GPT5.6-Luna ) was used in assisting with brainstorming and structuring base concept for this edit https://chatgpt.com/share/6a8ab5a8-8fc8-83ec-90e0-cfd35742e424</comment>
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      <text bytes="35701" sha1="g0x6u12lnmm2qzhd4nv28s4vvj6esvl" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013). The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the appraisals made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously. Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat. Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997). Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. 

lightly talk about some or all (?)

* individual differences such as self-esteem, rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. 

* self esteem
* previous relationship experiences
* rejection sensitivity
* personality tendances
* these things may increase sensitivity to threats or how situations are interpreted
* gender and culture (Zandbergen &amp; Brown, 2015) from social post 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
taken together, the evidence suggests that romantic jealousy is best understood as an interaction between relationship circumstances, perceived threat, cognitive appraisal and individual characteristics rather than just as the product of a signal cause.
*Different explanatory perspectives may operate at different levels of the same psychological process. Appraisal theory may explain how a situation becomes psychologically threatening. attachment theory helps explain why people may differ in their expectations and sensitivity to relationship threat. Individual differences explains why people facing similar circumstances may nevertheless respond differently.
*add figure 

* add learning feature
* feature box
* quiz: 

== What are the impacts of romantic jealousy? ==
Once romantic jealousy is experienced, it can influence how people think, feel and behave, with consequences that may extend beyond the individual to the relationship itself.

* connect back to last section
* say jealousy doesn't produce one universal outcome
* consequences depend partly on how jealousy is experiences and responded to (can have adaptive responses and harmful responses)
* briefly overview subheadings

(don't know if i need a source here, as i am being general, maybe i can just reuse a source about jealousy having impacts in general from above )

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship.

* cognitive jealousy is what you think/ suspect cite (Leite et al., 2024) 
* jealousy can heighten attention towards relationship information, people may be focused on factors mentioned above (rival, behavior, signs of rejection, evidence confirmation or contradiction)
* interpretation of ambiguous information (a partner talking closely with someone else does not automatically show romantic interest, but someone experiencing strong jealousy may interpret ambiguous information through the lens of a perceived threat. possible negative assumptions, suspicion, threat focused) cite (Brenner, 2018)
* repetitive thinking and information seeking (rumination, repeatedly thinking about threat, checking for evidence, mental replaying) reuse (Elphinston et al., 2013) for rumination 
* cognitive effects (effects on thoughts) 

explain that appraisal can continue after jealousy is experienced, with attention, interpretation, and rumination becoming focused on relationship relevant information, which can influence their emotional and behavioural responses. 

=== Emotional impacts ===
The emotional impact of romantic jealous can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. 

* jealousy can involve multiple emotional experiences (it can co-occur with other emotions, different people can experience this differently, and may experience multiple emotions at the same time) cite (Guerrero et al., 2005)
* emotional intensity can vary (the intensity of emotional experiences may be related to context, uncertainty, threat, individual differences discussed before) 
* emotional distress (intense or persistent jealousy may contribute to emotional distress) cite (Bush et al., 1988)
* emotion isn't automatically dysfunctional (unpleasant emotions can still serve psychological functions and draw attention to something important, therefore feeling jealousy does not automatically mean the person or relationship is unhealthy) cite (Attridge, 2013)

as established earlier, experiencing an unpleasant emotion does not necessarily make the emotion dysfunctional. 

=== Behavioural responses ===
The experience of jealousy does not create a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. 

* jealousy can motivate action (emotions can motivate behaviours, jealousy signals that relationship may be threatened so people may be motivated to respond.) cite (Fiske, 2002)
* as established earlier, jealousy does not determine a single behavioural respons
* constructive responses (jealousy can be followed by constructive behaviours link communicating feelings, discussing expectations, seeking reassurance, setting boundaries, and getting clarification. these constructive responses may allow someone to respond to an underlying concern without escalating the situation) maybe use (Attridge, 2013) again 
* less constructive responses ( MAY cause rumination, monitoring, accusations, withdrawal, testing partner, restrictive behaviour on partner.) cite (Yoshimura, 2004) 
* these behaviours can lead into more harmful behaviours.   

{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs withing an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. 

* communication (can prompt constructive typos and less constructive types. the effects depend partly on how the jealousy is communicated) 
* trust (jealousy related suspicion and monitory can be related to lower trust in relationships) can use source about what jealousy is, maybe (Karakoçoğlu &amp; Hasdağ, 2025) again
* relationship satisfaction (problematic jealousy with jealousy related behaviours and poorer relationship quality) cite  (Elphinston et al., 2013)  again

jealousy may prompt behaviours intended to maintain or protect eh relationship, or it may prompt behaviours that can harm trust and communication. therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed 

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <comment>wrote up integrated explanations   GenAi (ChatGPT, GPT5.6-Luna ) was used in assisting with brainstorming and structuring base concept for this edit https://chatgpt.com/share/6a8ab5a8-8fc8-83ec-90e0-cfd35742e424</comment>
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      <text bytes="36299" sha1="o6b3nngx4dy83swha9g18k7twhxmqvf" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013). The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the appraisals made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously. Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat. Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997). Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once romantic jealousy is experienced, it can influence how people think, feel and behave, with consequences that may extend beyond the individual to the relationship itself.

* connect back to last section
* say jealousy doesn't produce one universal outcome
* consequences depend partly on how jealousy is experiences and responded to (can have adaptive responses and harmful responses)
* briefly overview subheadings

(don't know if i need a source here, as i am being general, maybe i can just reuse a source about jealousy having impacts in general from above )

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship.

* cognitive jealousy is what you think/ suspect cite (Leite et al., 2024) 
* jealousy can heighten attention towards relationship information, people may be focused on factors mentioned above (rival, behavior, signs of rejection, evidence confirmation or contradiction)
* interpretation of ambiguous information (a partner talking closely with someone else does not automatically show romantic interest, but someone experiencing strong jealousy may interpret ambiguous information through the lens of a perceived threat. possible negative assumptions, suspicion, threat focused) cite (Brenner, 2018)
* repetitive thinking and information seeking (rumination, repeatedly thinking about threat, checking for evidence, mental replaying) reuse (Elphinston et al., 2013) for rumination 
* cognitive effects (effects on thoughts) 

explain that appraisal can continue after jealousy is experienced, with attention, interpretation, and rumination becoming focused on relationship relevant information, which can influence their emotional and behavioural responses. 

=== Emotional impacts ===
The emotional impact of romantic jealous can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. 

* jealousy can involve multiple emotional experiences (it can co-occur with other emotions, different people can experience this differently, and may experience multiple emotions at the same time) cite (Guerrero et al., 2005)
* emotional intensity can vary (the intensity of emotional experiences may be related to context, uncertainty, threat, individual differences discussed before) 
* emotional distress (intense or persistent jealousy may contribute to emotional distress) cite (Bush et al., 1988)
* emotion isn't automatically dysfunctional (unpleasant emotions can still serve psychological functions and draw attention to something important, therefore feeling jealousy does not automatically mean the person or relationship is unhealthy) cite (Attridge, 2013)

as established earlier, experiencing an unpleasant emotion does not necessarily make the emotion dysfunctional. 

=== Behavioural responses ===
The experience of jealousy does not create a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. 

* jealousy can motivate action (emotions can motivate behaviours, jealousy signals that relationship may be threatened so people may be motivated to respond.) cite (Fiske, 2002)
* as established earlier, jealousy does not determine a single behavioural respons
* constructive responses (jealousy can be followed by constructive behaviours link communicating feelings, discussing expectations, seeking reassurance, setting boundaries, and getting clarification. these constructive responses may allow someone to respond to an underlying concern without escalating the situation) maybe use (Attridge, 2013) again 
* less constructive responses ( MAY cause rumination, monitoring, accusations, withdrawal, testing partner, restrictive behaviour on partner.) cite (Yoshimura, 2004) 
* these behaviours can lead into more harmful behaviours.   

{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs withing an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. 

* communication (can prompt constructive typos and less constructive types. the effects depend partly on how the jealousy is communicated) 
* trust (jealousy related suspicion and monitory can be related to lower trust in relationships) can use source about what jealousy is, maybe (Karakoçoğlu &amp; Hasdağ, 2025) again
* relationship satisfaction (problematic jealousy with jealousy related behaviours and poorer relationship quality) cite  (Elphinston et al., 2013)  again

jealousy may prompt behaviours intended to maintain or protect eh relationship, or it may prompt behaviours that can harm trust and communication. therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed 

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36215" sha1="htro4s9ibmfy334s1ujbpspiz45zw1g" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013). The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously. Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat. Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997). Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship.

* cognitive jealousy is what you think/ suspect cite (Leite et al., 2024) 
* jealousy can heighten attention towards relationship information, people may be focused on factors mentioned above (rival, behavior, signs of rejection, evidence confirmation or contradiction)
* interpretation of ambiguous information (a partner talking closely with someone else does not automatically show romantic interest, but someone experiencing strong jealousy may interpret ambiguous information through the lens of a perceived threat. possible negative assumptions, suspicion, threat focused) cite (Brenner, 2018)
* repetitive thinking and information seeking (rumination, repeatedly thinking about threat, checking for evidence, mental replaying) reuse (Elphinston et al., 2013) for rumination 
* cognitive effects (effects on thoughts) 

explain that appraisal can continue after jealousy is experienced, with attention, interpretation, and rumination becoming focused on relationship relevant information, which can influence their emotional and behavioural responses. 

=== Emotional impacts ===
The emotional impact of romantic jealous can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. 

* jealousy can involve multiple emotional experiences (it can co-occur with other emotions, different people can experience this differently, and may experience multiple emotions at the same time) cite (Guerrero et al., 2005)
* emotional intensity can vary (the intensity of emotional experiences may be related to context, uncertainty, threat, individual differences discussed before) 
* emotional distress (intense or persistent jealousy may contribute to emotional distress) cite (Bush et al., 1988)
* emotion isn't automatically dysfunctional (unpleasant emotions can still serve psychological functions and draw attention to something important, therefore feeling jealousy does not automatically mean the person or relationship is unhealthy) cite (Attridge, 2013)

as established earlier, experiencing an unpleasant emotion does not necessarily make the emotion dysfunctional. 

=== Behavioural responses ===
The experience of jealousy does not create a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. 

* jealousy can motivate action (emotions can motivate behaviours, jealousy signals that relationship may be threatened so people may be motivated to respond.) cite (Fiske, 2002)
* as established earlier, jealousy does not determine a single behavioural respons
* constructive responses (jealousy can be followed by constructive behaviours link communicating feelings, discussing expectations, seeking reassurance, setting boundaries, and getting clarification. these constructive responses may allow someone to respond to an underlying concern without escalating the situation) maybe use (Attridge, 2013) again 
* less constructive responses ( MAY cause rumination, monitoring, accusations, withdrawal, testing partner, restrictive behaviour on partner.) cite (Yoshimura, 2004) 
* these behaviours can lead into more harmful behaviours.   

{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs withing an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. 

* communication (can prompt constructive typos and less constructive types. the effects depend partly on how the jealousy is communicated) 
* trust (jealousy related suspicion and monitory can be related to lower trust in relationships) can use source about what jealousy is, maybe (Karakoçoğlu &amp; Hasdağ, 2025) again
* relationship satisfaction (problematic jealousy with jealousy related behaviours and poorer relationship quality) cite  (Elphinston et al., 2013)  again

jealousy may prompt behaviours intended to maintain or protect eh relationship, or it may prompt behaviours that can harm trust and communication. therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed 

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36254" sha1="6udbmq17pzo27zoizgwv6fnpb3m7p8j" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013). The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously. Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat. Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997). Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018). Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealous can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. 

* jealousy can involve multiple emotional experiences (it can co-occur with other emotions, different people can experience this differently, and may experience multiple emotions at the same time) cite (Guerrero et al., 2005)
* emotional intensity can vary (the intensity of emotional experiences may be related to context, uncertainty, threat, individual differences discussed before) 
* emotional distress (intense or persistent jealousy may contribute to emotional distress) cite (Bush et al., 1988)
* emotion isn't automatically dysfunctional (unpleasant emotions can still serve psychological functions and draw attention to something important, therefore feeling jealousy does not automatically mean the person or relationship is unhealthy) cite (Attridge, 2013)

as established earlier, experiencing an unpleasant emotion does not necessarily make the emotion dysfunctional. 

=== Behavioural responses ===
The experience of jealousy does not create a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. 

* jealousy can motivate action (emotions can motivate behaviours, jealousy signals that relationship may be threatened so people may be motivated to respond.) cite (Fiske, 2002)
* as established earlier, jealousy does not determine a single behavioural respons
* constructive responses (jealousy can be followed by constructive behaviours link communicating feelings, discussing expectations, seeking reassurance, setting boundaries, and getting clarification. these constructive responses may allow someone to respond to an underlying concern without escalating the situation) maybe use (Attridge, 2013) again 
* less constructive responses ( MAY cause rumination, monitoring, accusations, withdrawal, testing partner, restrictive behaviour on partner.) cite (Yoshimura, 2004) 
* these behaviours can lead into more harmful behaviours.   

{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs withing an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. 

* communication (can prompt constructive typos and less constructive types. the effects depend partly on how the jealousy is communicated) 
* trust (jealousy related suspicion and monitory can be related to lower trust in relationships) can use source about what jealousy is, maybe (Karakoçoğlu &amp; Hasdağ, 2025) again
* relationship satisfaction (problematic jealousy with jealousy related behaviours and poorer relationship quality) cite  (Elphinston et al., 2013)  again

jealousy may prompt behaviours intended to maintain or protect eh relationship, or it may prompt behaviours that can harm trust and communication. therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed 

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36514" sha1="83qa9lb1ic15ddov7knx8dsgnevuv9a" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013). The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously. Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat. Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997). Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018). Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealousy can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. Jealousy can co-occur with multiple emotional experiences, and these different emotions can influence how jealousy is acted upon. For example, it has been found that people who experience jealousy alongside hostility but no guilt may be more likely to engage in violent behaviour. At the same time, high levels of passion can influence the desire to confront a rival (Guerrero et al., 2005). The emotional intensity of jealousy can vary by context and shape how people feel about themselves and the situation. Research has found that higher levels of threat connected to jealousy result in greater surprise, fear, and distress (Bush et al., 1988), showing that jealousy does, in fact, affect emotions. 

Despite jealousy's emotional impact, the emotions felt are not automatically dysfunctional or negative, as unpleasant emotions can sometimes serve a psychological function by drawing attention to something important. Attridge (2013) reflected this when stating that romantic jealousy can &quot;act as an adaptive emotion that is necessary to aid those who are in danger of losing their relationship, and must act to prevent the potential loss&quot;.

=== Behavioural responses ===
The experience of jealousy does not create a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. 

* jealousy can motivate action (emotions can motivate behaviours, jealousy signals that relationship may be threatened so people may be motivated to respond.) cite (Fiske, 2002)
* as established earlier, jealousy does not determine a single behavioural respons
* constructive responses (jealousy can be followed by constructive behaviours link communicating feelings, discussing expectations, seeking reassurance, setting boundaries, and getting clarification. these constructive responses may allow someone to respond to an underlying concern without escalating the situation) maybe use (Attridge, 2013) again 
* less constructive responses ( MAY cause rumination, monitoring, accusations, withdrawal, testing partner, restrictive behaviour on partner.) cite (Yoshimura, 2004) 
* these behaviours can lead into more harmful behaviours.   

{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs withing an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. 

* communication (can prompt constructive typos and less constructive types. the effects depend partly on how the jealousy is communicated) 
* trust (jealousy related suspicion and monitory can be related to lower trust in relationships) can use source about what jealousy is, maybe (Karakoçoğlu &amp; Hasdağ, 2025) again
* relationship satisfaction (problematic jealousy with jealousy related behaviours and poorer relationship quality) cite  (Elphinston et al., 2013)  again

jealousy may prompt behaviours intended to maintain or protect eh relationship, or it may prompt behaviours that can harm trust and communication. therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed 

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36414" sha1="9ffafqj6gqfc1p15jrkiv2z0qdhw88d" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013).   

The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously.

Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat.  

Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997).  

Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. 

A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. 

Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018).

Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealousy can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. Jealousy can co-occur with multiple emotional experiences, and these different emotions can influence how jealousy is acted upon. For example, it has been found that people who experience jealousy alongside hostility but no guilt may be more likely to engage in violent behaviour. At the same time, high levels of passion can influence the desire to confront a rival (Guerrero et al., 2005). The emotional intensity of jealousy can vary by context and shape how people feel about themselves and the situation. Research has found that higher levels of threat connected to jealousy result in greater surprise, fear, and distress (Bush et al., 1988), showing that jealousy does, in fact, affect emotions. 

Despite jealousy's emotional impact, the emotions felt are not automatically dysfunctional or negative, as unpleasant emotions can sometimes serve a psychological function by drawing attention to something important. Attridge (2013) reflected this when stating that romantic jealousy can &quot;act as an adaptive emotion that is necessary to aid those who are in danger of losing their relationship, and must act to prevent the potential loss&quot;.

=== Behavioural responses ===
Jealousy does not produce a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. Fiske (2022) discusses how emotions can motivate actions, especially based on their history and situation, and this is true for jealousy. 

As established earlier, jealousy does not determine a single behavioural response. One example of a behavioural response can have to do with constructive responses, like fixing expectations, seeking assurance, setting boundaries, and getting clarification, all things which can allow someone to respond in an adaptive way (Attridge, 2013).

Less constructive responses can include suspicious behaviour, negative communication, manipulation, or aggression (Yoshimura, 2004). These behaviours can spiral into further negativity, negatively affecting the individual involved and the relationship as a whole.{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs withing an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. 

* communication (can prompt constructive typos and less constructive types. the effects depend partly on how the jealousy is communicated) 
* trust (jealousy related suspicion and monitory can be related to lower trust in relationships) can use source about what jealousy is, maybe (Karakoçoğlu &amp; Hasdağ, 2025) again
* relationship satisfaction (problematic jealousy with jealousy related behaviours and poorer relationship quality) cite  (Elphinston et al., 2013)  again

jealousy may prompt behaviours intended to maintain or protect eh relationship, or it may prompt behaviours that can harm trust and communication. therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed 

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36512" sha1="kjy19x7re5a78g60fdah309q0pqsdcw" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013).   

The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously.

Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat.  

Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997).  

Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. 

A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. 

Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018).

Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealousy can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. Jealousy can co-occur with multiple emotional experiences, and these different emotions can influence how jealousy is acted upon. For example, it has been found that people who experience jealousy alongside hostility but no guilt may be more likely to engage in violent behaviour. At the same time, high levels of passion can influence the desire to confront a rival (Guerrero et al., 2005). The emotional intensity of jealousy can vary by context and shape how people feel about themselves and the situation. Research has found that higher levels of threat connected to jealousy result in greater surprise, fear, and distress (Bush et al., 1988), showing that jealousy does, in fact, affect emotions. 

Despite jealousy's emotional impact, the emotions felt are not automatically dysfunctional or negative, as unpleasant emotions can sometimes serve a psychological function by drawing attention to something important. Attridge (2013) reflected this when stating that romantic jealousy can &quot;act as an adaptive emotion that is necessary to aid those who are in danger of losing their relationship, and must act to prevent the potential loss&quot;.

=== Behavioural responses ===
Jealousy does not produce a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. Fiske (2022) discusses how emotions can motivate actions, especially based on their history and situation, and this is true for jealousy. 

As established earlier, jealousy does not determine a single behavioural response. One example of a behavioural response can have to do with constructive responses, like fixing expectations, seeking assurance, setting boundaries, and getting clarification, all things which can allow someone to respond in an adaptive way (Attridge, 2013).

Less constructive responses can include suspicious behaviour, negative communication, manipulation, or aggression (Yoshimura, 2004). These behaviours can spiral into further negativity, negatively affecting the individual involved and the relationship as a whole.{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs within an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. Jealousy can prompt communication that can be both constructive and less constructive. The effects jealousy has on a relationship depend partly on how it is communicated. 

Trust within the relationship can be affected, as jealousy-related suspicion and monitoring has been linked to lower trust and relationship dissatisfaction (Karakoçoğlu &amp; Hasdağ, 2025). In this context, overall relationship satisfaction can be negatively affected, as jealousy-related behaviours can become problematic and harm both the individual experiencing jealousy and their partner (Elphinston et al., 2013).

Jealousy may prompt behaviours intended to maintain or protect a relationship, or behaviours that can harm trust and communication. Therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed.

=== When can jealousy be harmful? ===
romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. 

* intensity cite maybe (Bush et al., 1988) (response proportionate to situation and if it overwhelms a persons ability to respond effectively, can determine if jealousy can be harmful)
* persistence (does jealousy resolve after the concern is address or does it continue, and does the person remain preoccupied with perceived threat) 
* behavioural expression (the presence of jealousy is not enough to label a relationship unhealthy, but the behavioural expression and consequences of jealousy are important factors to investigate. 
* impact on wellbeing (emotional distress, conflict, loss of trust, harm to both partners) cite (Mindful Health Solutions, 2023)
* jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression, and consequences. 

effective management of jealousy should not necessarily involve eliminating the emotion, but it may involve recognising the emotion, evaluating the perceived threat, and choosing responses that support individual and relationship wellbeing. 

== How can romantic jealousy be managed? ==
managing romantic jealousy involves more than suppressing an uncomfortable emotion. individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. jealousy can produce multiple emotional and behavioural responses, therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36399" sha1="q6cemtdl33pilgw85dvwe92sucbhgwi" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013).   

The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously.

Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat.  

Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997).  

Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. 

A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. 

Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018).

Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealousy can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. Jealousy can co-occur with multiple emotional experiences, and these different emotions can influence how jealousy is acted upon. For example, it has been found that people who experience jealousy alongside hostility but no guilt may be more likely to engage in violent behaviour. At the same time, high levels of passion can influence the desire to confront a rival (Guerrero et al., 2005). The emotional intensity of jealousy can vary by context and shape how people feel about themselves and the situation. Research has found that higher levels of threat connected to jealousy result in greater surprise, fear, and distress (Bush et al., 1988), showing that jealousy does, in fact, affect emotions. 

Despite jealousy's emotional impact, the emotions felt are not automatically dysfunctional or negative, as unpleasant emotions can sometimes serve a psychological function by drawing attention to something important. Attridge (2013) reflected this when stating that romantic jealousy can &quot;act as an adaptive emotion that is necessary to aid those who are in danger of losing their relationship, and must act to prevent the potential loss&quot;.

=== Behavioural responses ===
Jealousy does not produce a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. Fiske (2022) discusses how emotions can motivate actions, especially based on their history and situation, and this is true for jealousy. 

As established earlier, jealousy does not determine a single behavioural response. One example of a behavioural response can have to do with constructive responses, like fixing expectations, seeking assurance, setting boundaries, and getting clarification, all things which can allow someone to respond in an adaptive way (Attridge, 2013).

Less constructive responses can include suspicious behaviour, negative communication, manipulation, or aggression (Yoshimura, 2004). These behaviours can spiral into further negativity, negatively affecting the individual involved and the relationship as a whole.{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs within an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. Jealousy can prompt communication that can be both constructive and less constructive. The effects jealousy has on a relationship depend partly on how it is communicated. 

Trust within the relationship can be affected, as jealousy-related suspicion and monitoring has been linked to lower trust and relationship dissatisfaction (Karakoçoğlu &amp; Hasdağ, 2025). In this context, overall relationship satisfaction can be negatively affected, as jealousy-related behaviours can become problematic and harm both the individual experiencing jealousy and their partner (Elphinston et al., 2013).

Jealousy may prompt behaviours intended to maintain or protect a relationship, or behaviours that can harm trust and communication. Therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed.

=== When can jealousy be harmful? ===
Romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. The intensity of the jealousy can influence behaviour; if it overwhelms a person's ability to respond effectively, it can determine whether jealousy becomes more or less harmful (Bush et al., 1988). 

The persistence of jealousy can influence how harmful it is; one question is whether it resolves after the concern is addressed or continues, keeping an individual preoccupied with a perceived threat. This persistence can then influence behavioural expression. Jealousy alone is not enough to label a relationship as unhealthy or harmful, but the behavioural expression and consequences of jealousy are important factors to investigate.

Jealousy can also harm well-being, potentially causing emotional distress, conflict, loss of trust, and harm to both partners (Mindful Health Solutions, 2023). Jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression and consequences.

== How can romantic jealousy be managed? ==
Managing romantic jealousy involves more than suppressing an uncomfortable emotion. Individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. Jealousy can produce multiple emotional and behavioural responses; therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
the first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. 

* identify the emotion (jelaosy may come with a mixture of uncomfortable emotions. the person needs to be able to recognise tha thtey are feeling jealous adn but not that it must mean that their partner is doing something wrong) cite (Szczygieł et al., 2012)
* validate the emotion without validating the interpretation (acknowledging the emotion is real does not mean assuming that the interpretation producing it is accurate. for example, Maya can genuinely feel jealous without that meaning that Alex has actually been unfaithful 
* cite (Leahy &amp; Tirch, 2008) distinguishes between experiencing jealousy and acting on it 

=== Evaluating the perceived threat ===
after recognising jealousy, individuals can evaluate whether the perceived threat is supported by available evidence rather than assuming that the emotional response itself confirms the threat.  
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situation]]
* cognitive appraisal (acknowledge cognitve appraisal and bring back information from previous section, the person can ask themself what why and how things are happening in their mental processes) appraisal source: (Tomaka et al., 1997) and (McEachrane, 2009)
* distinguish evidence from assumption (figure 3.  these are three different things, and jealousy can become more difficult to manage when interpretations and predictions are treated as facts.
* consider uncertainty (sometimes there genuinely isn't enough information, so management may include tolerating some uncertainty rather than attempting to eliminate it through repeated checking or reassurance seeking) bring back source with  (Brenner, 2018) about uncertainty bring jealousy 
* avoid turning evaluation into surveillance (healthy evaluation involves reflection and open communication but does not mean checking phones, monitoring, tracking, or interrogating) 

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36476" sha1="6k33txkcacy02hrbq3e3tdloirvy241" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013).   

The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously.

Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat.  

Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997).  

Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. 

A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. 

Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018).

Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealousy can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. Jealousy can co-occur with multiple emotional experiences, and these different emotions can influence how jealousy is acted upon. For example, it has been found that people who experience jealousy alongside hostility but no guilt may be more likely to engage in violent behaviour. At the same time, high levels of passion can influence the desire to confront a rival (Guerrero et al., 2005). The emotional intensity of jealousy can vary by context and shape how people feel about themselves and the situation. Research has found that higher levels of threat connected to jealousy result in greater surprise, fear, and distress (Bush et al., 1988), showing that jealousy does, in fact, affect emotions. 

Despite jealousy's emotional impact, the emotions felt are not automatically dysfunctional or negative, as unpleasant emotions can sometimes serve a psychological function by drawing attention to something important. Attridge (2013) reflected this when stating that romantic jealousy can &quot;act as an adaptive emotion that is necessary to aid those who are in danger of losing their relationship, and must act to prevent the potential loss&quot;.

=== Behavioural responses ===
Jealousy does not produce a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. Fiske (2022) discusses how emotions can motivate actions, especially based on their history and situation, and this is true for jealousy. 

As established earlier, jealousy does not determine a single behavioural response. One example of a behavioural response can have to do with constructive responses, like fixing expectations, seeking assurance, setting boundaries, and getting clarification, all things which can allow someone to respond in an adaptive way (Attridge, 2013).

Less constructive responses can include suspicious behaviour, negative communication, manipulation, or aggression (Yoshimura, 2004). These behaviours can spiral into further negativity, negatively affecting the individual involved and the relationship as a whole.{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs within an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. Jealousy can prompt communication that can be both constructive and less constructive. The effects jealousy has on a relationship depend partly on how it is communicated. 

Trust within the relationship can be affected, as jealousy-related suspicion and monitoring has been linked to lower trust and relationship dissatisfaction (Karakoçoğlu &amp; Hasdağ, 2025). In this context, overall relationship satisfaction can be negatively affected, as jealousy-related behaviours can become problematic and harm both the individual experiencing jealousy and their partner (Elphinston et al., 2013).

Jealousy may prompt behaviours intended to maintain or protect a relationship, or behaviours that can harm trust and communication. Therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed.

=== When can jealousy be harmful? ===
Romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. The intensity of the jealousy can influence behaviour; if it overwhelms a person's ability to respond effectively, it can determine whether jealousy becomes more or less harmful (Bush et al., 1988). 

The persistence of jealousy can influence how harmful it is; one question is whether it resolves after the concern is addressed or continues, keeping an individual preoccupied with a perceived threat. This persistence can then influence behavioural expression. Jealousy alone is not enough to label a relationship as unhealthy or harmful, but the behavioural expression and consequences of jealousy are important factors to investigate.

Jealousy can also harm well-being, potentially causing emotional distress, conflict, loss of trust, and harm to both partners (Mindful Health Solutions, 2023). Jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression and consequences.

== How can romantic jealousy be managed? ==
Managing romantic jealousy involves more than suppressing an uncomfortable emotion. Individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. Jealousy can produce multiple emotional and behavioural responses; therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
The first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. Jealousy can be an uncomfortable and often negative emotion; however, recognising what you are feeling matters, as Szczygieł et al. (2012) notes that high emotional awareness can prevent the potentially damaging impact of negative emotions. 

It is also important to validate the emotion being felt without validating the interpretation, and to normalise feeling jealous while distinguishing between feeling and acting on the feeling (Leahy &amp; Tirch, 2008).

Maya can genuinely feel jealous based on her interpretation of Alex's behaviour, believing that Alex has actually been unfaithful.

=== Evaluating the perceived threat ===
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situati]]After recognising jealousy, individuals can evaluate whether available evidence supports the perceived threat rather than assuming the emotional response itself confirms it. Cognitive appraisal is important in managing jealousy, as recognising your mental process can help you understand how jealousy influences behaviour and ask yourself what, why, and how things are happening internally (McEachrane, 2009; Tomaka et al., 1997).

When evaluating threat, it helps to distinguish evidence from assumptions. See Figure 3 for the different elements of observation, interpretation, and prediction. These are not facts, and jealousy can be hard to manage when interpretations and predictions are treated as facts.

It is also important to consider the uncertainty. Sometimes there genuinely is not enough information, so managing jealousy may mean building tolerance for uncertainty rather than trying to eliminate it, since stress around the ambiguity of actions gives jealousy power (Brenner, 2018). To prevent jealousy from worsening, it is important to avoid turning evaluating an event into surveillance. Healthy evaluation involves reflection and open communication, but does not mean checking phones, monitoring, tracking, or interrogating.

=== Communication and relationship management ===
when jealousy concerns the relationship itself, constructive communication can allow partners to clarify expectations, discuss boundaries and address concerns without treating jealousy as justification for controlling behaviour. 

* communicate emotion not accuse (an accusation looks like 'you are obviously flirting with her' proper communication looks like ' i noticed that i felt uncomfortable and insecure when i saw that interaction and id like to talk about it' important to communicate the experience and avoid assuming intent)
* discuss relationship boundaries (partners may have different expectations about friendships, flirting, social media, and exclusivity. there isn't one universal relationship boundary. relationship management involves negotiating expectations rather than assuming that partners automatically share the same rules. 
* communication should not become control (feeling jealous does not create an entitlement to restrict a partners autonomy) cite (Young et al., 2019) 
* maya can communicate ' i feel threatened by what i saw, can we talk about what our expectations are' 

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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[[Category:Motivation and emotion/Book/Jealousy]]</text>
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      <text bytes="36700" sha1="a6quinif0c1wxkfhxynuu2b75luzlhc" xml:space="preserve"> {{title|Romantic jealousy:&lt;br&gt;Why does romantic jealousy occur, what are it's impacts, and how can It be managed?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

[[File:Man using smartphone outdoors.jpg|Man_using_smartphone_outdoors|right|thumb|150px|'''Figure 1'''. Person viewing an ambiguous message on a phone]]

;  Scenario

Maya has been in a committed romantic relationship for 3 years.

Her partner, Alex, has recently started spending more time with a new colleague, Joan.

One evening, Maya notices that Alex has been messaging Joan more often. She sees a message that, while not showing clear evidence of infidelity, could be interpreted in multiple ways.

Maya suddenly feels jealous.

She starts thinking: '' Why are they messaging so much? Is there something going on? Does Alex find Joan more interesting than me? Am I overreacting? '' 

Maya feels anxious and has an urge to check Alex's messages or ask what is happening.

Maya's situation shows how an ambiguous event can become psychologically significant when interpreted as a potential threat (see Figure 1).

What is happening to Maya, and what should she do with this feeling?
{{RoundBoxBottom}}

Romantic jealousy is an emotional response that can arise when a person perceives a threat to a valued romantic relationship. The experience and consequences of jealousy can vary substantially across people and situations. Romantic [[wikipedia:Jealousy|Jealousy]] involves an emotional response to a perceived threat to a valued romantic relationship. The threat can be real or imagined, perceived through the presence of a rival, or felt through fear of loss (Karakoçoğlu &amp; Hasdağ, 2025). In jealousy, [[wikipedia:Interpretation_(logic)|interpretation]] and [[wikipedia:Perception|perception]] play important roles. 

The onset and experiences of romantic jealousy can vary significantly between individuals because individual differences (such as gender and sexual orientation) and social differences (such as relationship status and type) shape how people interact and respond in any given situation (Valentova et al., 2020).

Jealousy can influence thoughts, emotions, and behaviour, and it is often associated with [[wikipedia:Cognition|cognition]] (Guerrero &amp; Andersen, 1996). The feeling of jealousy alone does not necessarily determine behaviour; however, the different responses drawn from the emotion can have different consequences for romantic relationships, with romantic jealousy often being tied to damaging effects (Elphinston et al., 2013)

Understanding jealousy is important for managing the emotion and its consequences, as it can help people distinguish emotional reactions from assumptions and actions. Understanding, attention, regulation, and constructing responses are all important steps in behaviour change (Duckworth &amp; Gross, 2020) and can help manage jealousy.{{RoundBoxTop|theme=1}}
'''Focus questions'''

* What is romantic jealousy, and how can it be understood as an emotional and motivational response?
* Why does romantic jealousy occur, and why might people experience it differently?
* What are the cognitive, emotional, behavioural, and relationship impacts of romantic jealousy?
* How can romantic jealousy be managed in ways that support healthy functioning?
{{RoundBoxBottom}}

== What is romantic jealousy? ==
Romantic jealousy is a complex psychological response to a perceived threat to a valued romantic relationship, involving interacting emotional, cognitive, and motivational processes, including [[wikipedia:Attachment_theory|attachment]] and behaviour (Sharpsteen &amp; Kirkpatrick, 1997). Jealousy is not just a single feeling felt in response to one particular event. It can be an emotion that is tied to other feelings in response to many scenarios. Individuals can interpret similar situations differently; thus, jealousy is unique to each person. It is important to understand what jealousy is, how it connects to other emotions, and the influence it can have on an individual or relationship. 

=== Defining romantic jealousy ===
As previously outlined, jealousy is a psychological response that occurs when a person perceives a threat to an important relationship. Romantic jealousy is a subtype of jealousy that concerns individuals in a romantic or intimate relationship. In a romantic relationship, the bond between the individuals involved is what is threatened, whether by a potential or imagined loss (suspicious jealousy) or an actual loss (fait accompli jealousy) (Miller &amp; Benz, 2013).   

The situation in which the perceived threat is felt does not have to involve an objectively confirmed threat. Differences between events and situations that incite jealousy may reflect the [[wikipedia:Appraisal_theory|appraisals]] made about the situation and how those appraisals elicit emotions (Scherer et al., 2001). The experience of jealousy may involve cognition, emotion, and motivation. Behaviour may follow these factors directly; however, it is not necessary and is not the same as experiencing jealousy.  

Maya's reaction of feeling jealous illustrates how the meaning attributed to an ambiguous event can contribute to jealousy. 

=== Jealousy and related emotional experiences ===
Although romantic jealousy can involve emotions such as [[wikipedia:Fear|fear]], [[wikipedia:Anxiety|anxiety]], [[wikipedia:Anger|anger]], and [[wikipedia:Insecurity|insecurity]], it should be distinguished from related psychological experiences such as envy. Jealousy aligns with the above-given definition of a valued relationship plus a perceived threat involving another person. Envy is separate and distinct. Envy is generally a negative emotion associated with wanting or desiring what someone else has. Envy can involve anything, such as a physical item, wealth, success, power, or a trait or quality. A distinction between jealousy and envy, as demonstrated in Table 1, is that jealousy is concerned with a threat or loss, whilst envy is concerned with wanting to gain something. Several other emotions may accompany jealousy; however, they are not interchangeable, as jealousy is its own unique emotion.  
  
{| class=&quot;wikitable&quot;
|+Table 1. Distinguishing Romantic Jealousy from Related Experiences 
!Experience
!Central Concern
!Example
|-
|Romantic Jealousy 
|Perceived threat to a valued romantic relationship
|Partner appears unusually close to another person 
|-
|Envy
|Wanting something another person has
|Another person has a relationship you want 
|-
|Fear/Anxiety
|Anticipated or perceived threat/danger
|Worry that a relationship may end 
|}

=== Is jealousy inherently harmful? ===
Experiencing romantic jealousy does not necessarily mean you are experiencing a harmful relationship event or outcome. The significance of jealousy may depend partly on how you interpret the experience and how you express it. An important point to note is that emotions do not equal behaviour. Feeling jealous, or even imagining behaving based on that jealousy, does not mean an individual is acting jealously.

Jealousy is not always irrational or out of the blue; often, it stems from somewhere, so the emotion can have benefits. Dillon (2013) wrote that &quot; beginning in infancy, humans display reactivity toward potential threats that can affect their care which is connected to their ability to thrive&quot; and that &quot;reactivity in the form of jealousy, serves to regain attention and care&quot;. This shows that jealousy can have an adaptive or positive function, potentially acting as an information signal, motivating attention toward a relationship concern, motivating relationship-protective behaviour, or prompting communication. In contrast to jealousy's potential positive outcomes, it can also prompt maladaptive responses, including conflict, suspicion, and controlling behaviour (Elphinston et al., 2013).  &lt;quiz display=&quot;simple&quot;&gt;
{Which statement best describes romantic jealousy?:
|type=&quot;()&quot;}


- It only occurs when a partner has definitely been unfaithful. 
- It is the same as envy. 
+ It can involve a perceived threat to a valued romantic relationship and may involve emotional, cognitive, and motivational processes. 
- It always results in harmful behaviour.
&lt;/quiz&gt;

== Why does romantic jealousy occur? ==
Romantic jealousy is unlikely to stem from relationship events alone. Instead, it can stem from the interaction among perceived threat, cognitive appraisal, relationship characteristics, and individual differences.  

=== Perceived relationship threat and cognitive appraisal ===
A central explanation for romantic jealousy is that situations become emotionally significant when people appraise them as threatening to a valued relationship. A potential threat is not automatically a psychological threat. A partner interacting with another person may not provoke jealousy, and the interaction becomes important only when interpreted as threatening (Martínez-León et al., 2017), which can involve many types of threat.  

Cognitive appraisal plays a central role in eliciting threat and challenge responses like jealousy, as it involves emotional judgements and assessments (McEachrane, 2009; Tomaka et al., 1997). Appraisals contribute to emotional responses by helping someone interpret what they are experiencing.   

=== Attachment and relationship security ===
Attachment theory offers one explanation for why individuals may differ in their sensitivity to relationship threat and how they experience and express jealousy. Humans are born with attachment behaviours that help ensure proximity to important figures who can provide protection (Shaver &amp; Mikulincer, 2008). Romantic relationships can function as important attachments; however, people differ in expectations about what attachments can look like, which can influence how they respond to relationship threat (Sharpsteen &amp; Kirkpatrick, 1997).  

Some people hold an anxious attachment style, which is defined as &quot;uncertainty regarding the availability of attachment figures&quot; (Campbell &amp; Marshall, 2011), which can increase sensitivity to loss or threat and make jealous behaviours stronger. Avoidant attachment style is defined as &quot;an active fear of closeness&quot; (Bartholomew, 1990), which can influence how jealousy is expressed or managed. 

Maya's attachment-related expectations could influence how sensitive she is to ambiguous relationships, particularly if she has an anxious attachment style.

=== Relationship context and uncertainty ===
Jealousy is also shaped by relationship characteristics and the immediate context, particularly when information about the relationship or a potential rival is uncertain. A study by Knobloch et al. (2001) states that &quot;relational uncertainty and intimacy are two indicators of relationship development that are likely to coincide with people's propensity to experience cognitive and emotional jealousy&quot; which supports that when people are uncertain about where their relationship stands, how committed their partner is, what boundaries exist or what a partners behaviour means, they may be more likely to get jealous. Uncertainty can make unclear information harder to interpret, making the information feel more impactful.

Relationship factors like intimacy, commitment, trust, relationship satisfaction, perceived exclusivity, and communication style shape jealousy. Additionally, the relationship context and perceptions can shape reactions. These contextual factors might include perceptions about rivals, previous breaches of trust, interactions, or types of communication taking place. The relationship context determines what information is available and how ambiguous that information might feel.

Any recent conflict, unclear boundaries, or uncertainty between Alex and Maya could make Alex's messages to Joan feel more threatening. Jealousy cannot be explained by internal traits alone; the relationship situation matters

=== Individual differences ===
Beyond attachment and relationship characteristics, individual differences can influence how strongly people perceive and respond to potential romantic threats. Individual differences such as rejection sensitivity, previous relationship experiences and personality may influence sensitivity to perceived relationship threat and responses to jealousy. These individual factors contribute to differences in how individuals interpret information and cannot necessarily be predicted. 

A 2015 study by Zandbergen &amp; Brown found that individual factors influence jealousy at different levels. One finding states that gender was a stronger predictor of jealousy concerning emotional cheating, while culture was a stronger predictor of jealousy concerning sexual cheating (Zandbergen &amp; Brown, 2015). Additionally, it was found that past experiences in connection with gender influence the type of jealousy that might be felt (Zandbergen &amp; Brown, 2015). This study supports the idea that both smaller and larger individual differences influence the way information is interpreted and how jealousy may be felt and acted upon. 

=== Integrated explanations ===
{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees the same message from Alex to Joan in two different circumstances

Scenario A:
Maya feels secure in her relationship and knows how Alex has openly discussed the friendship.

Scenario B: 
Maya and Alex recently experienced conflict and have not discussed relationship boundaries.{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Why might the same message produce different levels of jealousy?:
|type=&quot;()&quot;}


- The message automatically created jealousy.
- Jealousy depends only on personality.
+ The meaning of the message may be shaped by appraisal, relationship security and contextual uncertainty. 
- One person must be irrational.
&lt;/quiz&gt;
[[File:Relationship event.png|thumb|Figure 2. Model of romantic jealousy going from event, to appraisal, to emotional and behavioral responses]]
Taken together, the evidence suggests that romantic jealousy is best understood as an interaction among relationship circumstances, perceived threat, cognitive appraisal, and individual characteristics rather than as the product of a single cause. Different explanations for jealousy may operate at different levels of the same psychological process. 

Appraisal theory may explain how a situation becomes psychologically threatening, while attachment theory may explain why people differ in their expectations and sensitivity to relationship threat. Individual differences explain why people facing similar circumstances may respond differently. As shown in Figure 2, many factors separate a relationship event from the emotion of jealousy and how it is acted upon, with each factor coming in at different steps in the process of emotion and behaviour formation. 

== What are the impacts of romantic jealousy? ==
Once people experience romantic jealousy, it can influence how they think, feel, and behave, with consequences that may extend beyond the individual to the relationship itself. As noted in previous sections, jealousy is not felt the same way for the same reasons, so it will not produce one universal outcome. Consequences of jealousy depend partly on how it is experienced and responded to, with the potential for both adaptive and harmful responses.

=== Cognitive impacts ===
Romantic jealousy can influence cognitive processing by increasing attention to potential relationship threats and shaping how individuals interpret ambiguous information about their partner and relationship. Cognitive jealousy concerns what one thinks or suspects, and may, for instance, look like &quot;I am worried that someone of the opposite sex is stalking my partner&quot;(Leite et al., 2024). Jealousy can heighten attention towards relationship information cognitively and make someone hyperalert. This cognitive form of jealousy can affect how ambiguous information is interpreted, as someone experiencing strong jealousy may view it through the lens of negative assumptions and suspicion, with ambiguity having a stronger effect on the mind (Brenner, 2018).

Another cognitive impact of jealousy is how thoughts about the relationship are repeated to try to gain new information through checking for evidence and mental replaying. Rumination has been highlighted as a factor that can explain the link between romantic jealousy and relationship dissatisfaction (Elphinston et al., 2013). Threat appraisal can continue after jealousy is initially experienced, with attention, interpretation, and rumination focusing on relationship-relevant information, which can influence emotional and behavioural responses

=== Emotional impacts ===
The emotional impact of romantic jealousy can extend beyond the experience of jealousy itself, with individuals potentially experiencing anxiety, fear, anger, sadness, or insecurity in response to perceived relationship threat. Jealousy can co-occur with multiple emotional experiences, and these different emotions can influence how jealousy is acted upon. For example, it has been found that people who experience jealousy alongside hostility but no guilt may be more likely to engage in violent behaviour. At the same time, high levels of passion can influence the desire to confront a rival (Guerrero et al., 2005). The emotional intensity of jealousy can vary by context and shape how people feel about themselves and the situation. Research has found that higher levels of threat connected to jealousy result in greater surprise, fear, and distress (Bush et al., 1988), showing that jealousy does, in fact, affect emotions. 

Despite jealousy's emotional impact, the emotions felt are not automatically dysfunctional or negative, as unpleasant emotions can sometimes serve a psychological function by drawing attention to something important. Attridge (2013) reflected this when stating that romantic jealousy can &quot;act as an adaptive emotion that is necessary to aid those who are in danger of losing their relationship, and must act to prevent the potential loss&quot;.

=== Behavioural responses ===
Jealousy does not produce a single behavioural response. Individuals may respond through constructive communication and relationship maintenance, or through behaviours that increase conflict, control or harm. Fiske (2022) discusses how emotions can motivate actions, especially based on their history and situation, and this is true for jealousy. 

As established earlier, jealousy does not determine a single behavioural response. One example of a behavioural response can have to do with constructive responses, like fixing expectations, seeking assurance, setting boundaries, and getting clarification, all things which can allow someone to respond in an adaptive way (Attridge, 2013).

Less constructive responses can include suspicious behaviour, negative communication, manipulation, or aggression (Yoshimura, 2004). These behaviours can spiral into further negativity, negatively affecting the individual involved and the relationship as a whole.{{RoundBoxTop|theme=3}}
;  Scenario

Maya sees Alex talking closely with Joan at a work party. She experiences jealousy. 

Response A:
Maya notices that she feels threatened, pauses before responding, and later tells Alex that the interaction makes her feel insecure. They discuss what happened. 

Response B: 
Maya becomes angry, repeatedly checks Alex's phone and accuses him of being unfaithful. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Which statement best explains the difference between Maya's two responses?:
|type=&quot;()&quot;}


- Jealousy was only present in response B.
- constructive responses require people to suppress jealousy.
+ The emotional experience is similar, while the behavioural response differs. 
- Jealousy always produces controlling behaviour.
&lt;/quiz&gt;

=== Relationship consequences ===
Because romantic jealousy occurs within an interpersonal relationship, its consequences can extend beyond the individual to influence communication, trust, satisfaction and relationship stability. Jealousy can prompt communication that can be both constructive and less constructive. The effects jealousy has on a relationship depend partly on how it is communicated. 

Trust within the relationship can be affected, as jealousy-related suspicion and monitoring has been linked to lower trust and relationship dissatisfaction (Karakoçoğlu &amp; Hasdağ, 2025). In this context, overall relationship satisfaction can be negatively affected, as jealousy-related behaviours can become problematic and harm both the individual experiencing jealousy and their partner (Elphinston et al., 2013).

Jealousy may prompt behaviours intended to maintain or protect a relationship, or behaviours that can harm trust and communication. Therefore, relationship consequences depend not only on whether jealousy occurs, but on how it is interpreted and expressed.

=== When can jealousy be harmful? ===
Romantic jealousy becomes particularly concerning when it is intense, persistent or difficult to regulate, especially when it is expressed through behaviours that undermine autonomy, wellbeing, trust, or relationship functioning. The intensity of the jealousy can influence behaviour; if it overwhelms a person's ability to respond effectively, it can determine whether jealousy becomes more or less harmful (Bush et al., 1988). 

The persistence of jealousy can influence how harmful it is; one question is whether it resolves after the concern is addressed or continues, keeping an individual preoccupied with a perceived threat. This persistence can then influence behavioural expression. Jealousy alone is not enough to label a relationship as unhealthy or harmful, but the behavioural expression and consequences of jealousy are important factors to investigate.

Jealousy can also harm well-being, potentially causing emotional distress, conflict, loss of trust, and harm to both partners (Mindful Health Solutions, 2023). Jealousy exists on a continuum, so problematic jealousy is better understood in terms of pattern, intensity, regulation, expression and consequences.

== How can romantic jealousy be managed? ==
Managing romantic jealousy involves more than suppressing an uncomfortable emotion. Individuals can benefit from recognising their emotional response, evaluating the perceived threat and choosing responses that support both personal wellbeing and relationship functioning. Jealousy can produce multiple emotional and behavioural responses; therefore, management needs to occur at several points in the process. 

=== Recognising the emotional response ===
The first step in managing jealousy is recognising and acknowledging the emotional response without automatically treating the feeling itself as evidence that the perceived threat is real. Jealousy can be an uncomfortable and often negative emotion; however, recognising what you are feeling matters, as Szczygieł et al. (2012) notes that high emotional awareness can prevent the potentially damaging impact of negative emotions. 

It is also important to validate the emotion being felt without validating the interpretation, and to normalise feeling jealous while distinguishing between feeling and acting on the feeling (Leahy &amp; Tirch, 2008).

Maya can genuinely feel jealous based on her interpretation of Alex's behaviour, believing that Alex has actually been unfaithful.

=== Evaluating the perceived threat ===
[[File:Relationship event (1).png|thumb|Figure 3. Maya's progression of thoughts on situati]]After recognising jealousy, individuals can evaluate whether available evidence supports the perceived threat rather than assuming the emotional response itself confirms it. Cognitive appraisal is important in managing jealousy, as recognising your mental process can help you understand how jealousy influences behaviour and ask yourself what, why, and how things are happening internally (McEachrane, 2009; Tomaka et al., 1997).

When evaluating threat, it helps to distinguish evidence from assumptions. See Figure 3 for the different elements of observation, interpretation, and prediction. These are not facts, and jealousy can be hard to manage when interpretations and predictions are treated as facts.

It is also important to consider the uncertainty. Sometimes there genuinely is not enough information, so managing jealousy may mean building tolerance for uncertainty rather than trying to eliminate it, since stress around the ambiguity of actions gives jealousy power (Brenner, 2018). To prevent jealousy from worsening, it is important to avoid turning evaluating an event into surveillance. Healthy evaluation involves reflection and open communication, but does not mean checking phones, monitoring, tracking, or interrogating.

=== Communication and relationship management ===
When jealousy concerns the relationship itself, constructive communication can help partners clarify expectations, discuss boundaries, and address concerns without treating jealousy as justification for controlling behaviour. Firstly, it is important to communicate an emotion rather than accuse. An accusation may look like &quot;you are obviously flirting with her&quot;. In contrast, proper communication may look like &quot;I noticed that I felt uncomfortable and insecure when I saw that interaction and I would like to talk about it&quot;. It is important to communicate the experience and avoid assuming intent. 

One way to combat jealousy is to discuss relationship boundaries. Partners may have different expectations about friendships, flirting, social media and exclusivity. No single relationship boundary fits everyone, and managing relationships may involve negotiating expectations rather than assuming partners automatically share the same rules.

Communication must not become control, as jealousy does not create an entitlement to restrict a partner's autonomy, and social control is associated with lower relationship satisfaction and more unhealthy transformations (Young et al., 2019).

In Maya's situation, Maya could communicate to Alex, saying, &quot;I feel threatened by what I saw, can we talk about what our expectations are&quot;

=== Emotion regulation ===
[[File:Emotional regulation flowchart.pdf|thumb|Figure 4. Emotional regulation flow chart]]
emotion regulation can help individuals manage the intensity of jealousy and its associated emotions, creating space to choose responses that are consistent with their longer term relationship goals.

* regulation is not suppression (suppression is 'i shouldn't feel this' regulation is 'i am feeling this but i can still decide how i respond) cite (Gross, 2008) 
* add mindfulness (mindfulness supports emotional awareness) (maybe don't keep or keep short)
* cognitive reappraisal (lightly reconnect to previous sections about rethinking meanings of situations cognitively) 
* behavioural pause (when emotional arousal is high, taking time before communicating can help prevent impulsive behaviour) Figure 4
* align regulation with relationship goals (what does the person ultimately want, potential goal is to protect the relationship, then which response is most likely to move toward that goal) 

=== When the threat is real ===
managing jealousy does not require dismissing every perceived threat as irrational. when evidence indicates a genuine relationship problem, appropriate management involves addressing the underlying issue rather than simply regulating the emotion away. 

* sometimes the perceived threat is real ( a partner could be violating a boundary, there may be infidelity, deception, repeated inappropriate behavior or a breakdown in trust) cite (Fincham &amp; May, 2017)
* emotional regulation doesn't mean tolerating mistreatment (if a persons jealousy is responding to genuine evidence of betrayal or boundary violation, simply telling them to 'manage their jealousy' could cause them to miss the underlying relationship problem)
* communicate and negotiate (if threat is real discuss what happened, clarify boundaries, communicate needs, decide what changes are necessary and determine if trust can be repaired,) cite (Abrahamson et al., 2011) 
* decide what is within ones control  (you cant control your partners choices, but you can control how you respond)
* professional support (when jealousy becomes persistent, severe, highly distressing, associated with controlling or aggressive behaviour, or is severely impairing relationship functioning, professional psychological support may help) 

effective management requires both emotional regulation and accurate threat detection.  

{{RoundBoxTop|theme=3}}
;  Scenario

Maya has talked to Alex about what she saw on his phone. 
Alex explains that Joan was just a work friend and there was no romantic relationship. 
{{RoundBoxBottom}}
&lt;quiz display=&quot;simple&quot;&gt;
{Maya still feels jealous; Which response best reflects the approach described above?:
|type=&quot;()&quot;}


- Suppress the jealousy because the threat has been disproven.
- Check Alex's phone until Maya feels completely certain.
+ Acknowledge the emotion, consider the evidence, tolerate some uncertainty and choose a constructive response. 
- Assume that feeling jealous must mean something is wrong. 
&lt;/quiz&gt;
==Conclusion==
* The Conclusion is arguably the most important section
* Draft clear take-home message(s), even at the topic development stage
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
* [[wikipedia:Cognitive_appraisal|Cognitive appraisal]] (Wikipedia)
* [[wikipedia:Emotional_self-regulation|Emotional self regulation]] (Wikipedia)
* [[Motivation and emotion/Book/2026/Love styles and relationships|Love styles and relationships]] (Wikiversity)
* [[wikipedia:Threat|Threat]] (Wikipedia)

==References ==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2011). What helps couples rebuild their relationship after infidelity? Journal of Family Issues, 33(11), 1494–1519. https://doi.org/10.1177/0192513x11424257

Attridge, M. (2013). Jealousy and relationship closeness. SAGE Open, 3(1), 215824401347605. https://doi.org/10.1177/2158244013476054

Bartholomew, K. (1990). Avoidance of Intimacy: An Attachment Perspective. Journal of Social and Personal Relationships, 7(2), 147–178. https://doi.org/10.1177/0265407590072001

Brenner, G. (2018, October 19). The six faces of jealousy {{!}} psychology today. Www.Psychologytoday.Com. https://www.psychologytoday.com/us/blog/experimentations/201810/the-six-faces-jealousy

Bush, C. R., Bush, J. P., &amp; Jennings, J. (1988). Effects of jealousy threats on relationship perceptions and emotions. Journal of Social and Personal Relationships, 5(3), 285–303. https://doi.org/10.1177/0265407588053002

Campbell, L., &amp; Marshall, T. (2011). Anxious attachment and relationship processes: An interactionist perspective. Journal of Personality, 79(6), 1219–1250. https://doi.org/10.1111/j.1467-6494.2011.00723.x

Dillon, L. (2013). Functional aspects of jealousy across the lifespan. Human Ethology, 28(2).

Duckworth, A. L., &amp; Gross, J. J. (2020). Behavior change. Organizational Behavior and Human Decision Processes, 161(1), 39–49. https://doi.org/10.1016/j.obhdp.2020.09.002

Elphinston, R. A., Feeney, J. A., Noller, P., Connor, J. P., &amp; Fitzgerald, J. (2013). Romantic jealousy and relationship satisfaction: The costs of rumination. Western Journal of Communication, 77(3), 293–304. https://doi.org/10.1080/10570314.2013.770161

Fincham, F. D, &amp; May, R. W. (2017). Infidelity in romantic relationships. Current Opinion in Psychology, 13(4), 70–74. https://doi.org/10.1016/j.copsyc.2016.03.008

Fiske, A. P. (2002). Socio-moral emotions motivate action to sustain relationships. Self and Identity, 1(2), 169–175. https://doi.org/10.1080/152988602317319357

Gross, J. J. (2008). Emotion regulation. In Handbook of emotions (pp. 497–513).

Guerrero, L. K., &amp; Andersen, P. A. (1996). Jealousy experience and expression in romantic relationships. Handbook of Communication and Emotion, 155–188. https://doi.org/10.1016/b978-012057770-5/50008-4Guerrero, L. K., Trost, M. R., &amp; Yoshimura, S. M. (2005). Romantic jealousy: Emotions and communicative responses. Personal Relationships, 12(2), 233–252. https://doi.org/10.1111/j.1350-4126.2005.00113.x

Karakoçoğlu, N., &amp; Hasdağ, D. (2025). Romantic jealousy: A comprehensive review. Psikiyatride Güncel Yaklaşımlar, 17(1), 179–196. https://doi.org/10.18863/pgy.1454542

KNOBLOCH, L. K., SOLOMON, D. H., &amp; CRUZ, M. G. (2001). The role of relationship development and attachment in the experience of romantic jealousy. Personal Relationships, 8(2), 205–224. https://doi.org/10.1111/j.1475-6811.2001.tb00036.x

Leahy, R. L., &amp; Tirch, D. D. (2008). Cognitive behavioral therapy for jealousy. International Journal of Cognitive Therapy, 1(1), 18–32. https://doi.org/10.1521/ijct.2008.1.1.18

Leite, Ã., Silva, B., Vilela, B., Rodrigues, I., Fernandes, J., Romão, J., &amp; Ribeiro, A. M. (2024). Measurement invariance of the multidimensional jealousy scale and quality of relationships inventory (friend). Behavioral Sciences, 14(1), 44. https://doi.org/10.3390/bs14010044

Martínez-León, N., Peña, J., Salazar, H., García, A., &amp; Sierra, J. (2017). APA PsycNet. In psycnet.apa.org. https://psycnet.apa.org/record/2017-47692-008

McEachrane, M. (2009). Emotion, meaning, and appraisal theory. Theory &amp; Psychology, 19(1), 33–53. https://doi.org/10.1177/0959354308101418

Miller, R. L., &amp; Benz, J. (2013). Jealousy, romantic. The Encyclopedia of Cross‐Cultural Psychology, 777–782. https://doi.org/10.1002/9781118339893.wbeccp311

Mindful Health Solutions. (2023, April 7). The deeper meaning of jealousy: A psychological exploration. Mindful Health Solutions. https://mindfulhealthsolutions.com/the-deeper-meaning-of-jealousy-a-psychological-exploration/

Scherer, K. R., Schorr, A., &amp; Johnstone, T. (2001). Appraisal processes in emotion. Oxford University Press, USA.

Sharpsteen, D. J., &amp; Kirkpatrick, L. A. (1997). Romantic jealousy and adult romantic attachment. Journal of Personality and Social Psychology, 72(3), 627–640. https://doi.org/10.1037/0022-3514.72.3.627

Shaver, P., &amp; Mikulincer, M. (2008). An overview of adult attachment theory. https://cheleyntema.com/wp-content/uploads/2024/05/Mikulincer-and-Shaver-2008-Overview-of-attachment-.pdf

Szczygieł, D., Buczny, J., &amp; Bazińska, R. (2012). Emotion regulation and emotional information processing: The moderating effect of emotional awareness. Personality and Individual Differences, 52(3), 433–437. https://doi.org/10.1016/j.paid.2011.11.005

Tomaka, J., Blascovich, J., Kibler, J., &amp; Ernst, J. M. (1997). Cognitive and physiological antecedents of threat and challenge appraisal. Journal of Personality and Social Psychology, 73(1), 63–72. https://doi.org/10.1037/0022-3514.73.1.63

Valentova, J. V., de Moraes, A. C., &amp; Varella, M. A. C. (2020). Gender, sexual orientation and type of relationship influence individual differences in jealousy: A large Brazilian sample. Personality and Individual Differences, 157, 109805. https://doi.org/10.1016/j.paid.2019.109805

Yoshimura, S. M. (2004). Emotional and behavioral responses to romantic jealousy expressions. Communication Reports, 17(2), 85–101. https://doi.org/10.1080/08934210409389378

Young, V. J., Burke, T. J., &amp; Curran, M. A. (2019). Interpersonal effects of health-related social control: Positive and negative influence, partner health transformations, and relationship quality. Journal of Social and Personal Relationships, 36(11–12), 3986–4004. https://doi.org/10.1177/0265407519846565

}}

==External links==
* [https://www.apa.org/topics/emotions Emotions] (American Psychological Association)
* [https://www.apa.org/topics/marriage-relationships/healthy-relationships Happy couples] (American Psychological Association)
* [https://services.unimelb.edu.au/counsel/resources/relationships/intimate-relationships Intimate relationships] (University of Melbourne)
* [https://www.apa.org/topics/marriage-relationships Marriage and relationships] (American Psychological Association)
* [https://www.apa.org/topics/physical-abuse-violence/relationships-dating-love-sex Violence in relationships] (American Psychological Association)

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      <text bytes="10681" sha1="0i7tsqetcc3ikoteg9taehz2yh4jmmt" xml:space="preserve">{{title|Developing a growth mindset:&lt;br&gt; How can a growth mindset be cultivated and sustained?}}
__TOC__
==Overview==
{{RoundBoxTop|theme=3}}
[[File:Openminded.png|right|thumb|150px|'''Figure 1'''. Illustration of being open-minded]]

; Imagine this ...

Two childhood best friends find themselves in two different places when they finish high school. Though living in the same neighbourhood, attending the same schools, enjoying the same hobbies and receiving equal opportunities, one is earning a scholarship to study their dream degree, and the other is falling behind. How could two people with the same opportunities have vastly separate life outcomes? Well, one evaded challenge for fear of failing, while the other appraised these failures as opportunities to learn. The margin between one and the other was not created based on ability, but from mindset. {{RoundBoxBottom}}

* A person's mindset is the foundation in which learning and motivation build up on. 
* Meaning the mindset that we adopt is highly influential in how we perceive our abilities and intelligence as well as how we conduct ourselves in facing new adversities
* In order to foster learning and motivation, an individual must shift from a fixed mindset to a growth mindset in order to embrace challenges and gain success
{{RoundBoxTop|theme=3}}
'''Focus questions'''

* How does a growth mindset differ from a fixed mindset?
* What is the role of neuroplasticity in changing and sustaining a mindset?
* How can failure, feedback and goal-setting affect the development of a growth mindset? 
* How do educational practices nurture a student's growth mindset?{{RoundBoxBottom}}

==Growth and Fixed Mindsets==

=== Mindsets ===

* Mindsets are an established process or set &quot;rules&quot; or structure that an individual adopts as part of their way of thinking, attitudes, and beliefs. 

* A mindset is highly influential in the way in which we conduct ourselves in terms of facing challenges, and how an individual may perceive failure and success (Dweck &amp; Leggett, 1988)

=== Growth versus fixed mindset's ===

* Mueller and Dweck (1988) established and developed two different Mindsets people adopt; growth and fixed, these mindsets run on a continuum
* Someone who adopts a growth mindset views that their abilities, skills and intelligence are sculpted, being able to grow and develop from setbacks as well as view failures as something to improve upon (Mueller &amp; Dweck, 1988)
* Someone who adopts a fixed mindset's see that their abilities, skills and intelligence is static and are simply as a result of a persons genetic disposition unlikely to change (Mueller &amp; Dweck, 1988)
* Implicit theory of intelligence - the belief that intelligence is an ability that can be shaped or &quot;moulded&quot; and developed (Dweck and Leggett, 1988)

== Physiological affects of mindset's ==

* The role of neuroplasticity is to enable and create new neural pathways throughout the brain especially important in learning a new skill or behaviour - proving that the ability of a person's mindset is not fixed but ever changing (e.g. the process of neurogenesis, or synaptogenesis)
* Therefore, repeated practice of a certain skill or behaviour (I.e. fixed Mindset), may strengthen these pathways in creating persistent and default processes in skill development - important in understanding how certain mindset's can be sustained
* In the context of growth Mindsets, neurobiology supports the underlying physiological mechanisms that demonstrate that a new mindset can be constructed on a neurological level to adapt to learning, and development (Sarrasin et al., 2018)

== Failure, feedback and goal setting == 

=== Responses to challenges and failure ===

* Learned helplessness theory – exposure to repeated negative outcomes or failures could lead to a person reducing their attempts to change a situation (Seligman &amp; Maier, 1967) 

* In the instance of Peter (see learning feature scenario), repeated failure reinforces the notion that their actions will not change their outcomes – this underlines the issue of how growth mindset is sustained or the beginning of a fixed mind/learned helplessness begins

=== Mastery goals versus performance goals ===
* Mastery goals involve an individual who is motivated in achieving competence, understanding and self-improvement (Dweck &amp; Leggett, 1988) 

* An individual motivated by performance goals seek to source their goals to appear competent relative to others 

* Dweck and Leggett (1988) connect goals to a growth mindset – the type of goals and motivation for goals may affect the types of mindset that you may adopt (Yu &amp; McLellan, 2020; Combette et al., 2024) 

== Mindset's in the educational setting ==

* Implicit theory of intelligence - the belief that intelligence is an ability that can be shaped or &quot;moulded&quot; and developed (Dweck and Leggett, 1988)

=== Pedagogical interventions and forming a growth mindset ===

*It is well versed that the intervention of praise and positive feedback from teachers and educators is a positive predictor through which mindset changes can be achieved (Gunderson et al., 2013)
*However, Mueller and Dweck (1998) suggested praise before failure may demonstrate lower persistence or high disappointment, but praise directed towards the effort of the student promotes improvement (person vs process praise)
*It is also important to consider the role of the teachers beliefs and attitudes in the classroom environment in which a growth mindset is successfully cultivated (Mesler et al., 2021)

=== Academic outcomes and a growth mindset ===

*Blackwell et al. (2007) suggested that the use of growth mindset interventions provided promising evidence in predicting that adopting growth mindset behaviours is reflective in positive academic achievement
*Also contested that a growth mindset was sufficient with its association with academic achievement by relating to mastery goals however has a weak association between goals and goal achievement (Burnette et al., 2018)
*Macnamara and Burgoyne (2023) systematic review and meta-analysis argue that such strong positive association between academic achievement and growth mindset interventions was as a result from targeting students with strong fixed mindsets


==Learning features==
{{Robelbox|title=Learned Helplessness scenario...|theme=3}}Peter has always been a good student in primary school. He always found that science was his strongest subject with his passion to become a biologist once he is older. Peter always takes pride in his abilities with his test results and assessment pieces always gaining praise from his teachers and parents.

When starting high school and feeling his most confident academically, he enrols in advanced science classes. However, he finds that high school science classes are far more difficult than he expected. After receiving a D in his first quiz and some harsh feedback, Peter feels discouraged and defeated. Peter believes that he is no longer good at science and believes he will never pass this class. As a result, Peter stops attending class and accepts that he will never become a biologist.{{RoundBoxBottom}} 

&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==


==See also==
* [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] (Book chapter, 2019)
* [[w:Mindset|Mindsets]] (Wikipedia article)


==References==

{{Hanging indent|1=

Blackwell, L. S., Trzesniewski, K. H., &amp; Dweck, C. S. (2007). Implicit Theories of Intelligence Predict Achievement Across an Adolescent Transition: A Longitudinal Study and an Intervention. Child Development, 78(1), 246–263. https://doi.org/10.1111/j.1467-8624.2007.00995.x

Burnette, J. L., Billingsley, J., Banks, G. C., Knouse, L. E., Hoyt, C. L., Pollack, J. M., &amp; Simon, S. (2023). A systematic review and meta-analysis of growth mindset interventions: For whom, how, and why might such interventions work? Psychological Bulletin, 149(3–4), 174–205. https://doi.org/10.1037/bul0000368

Combette, L. T., Rotgé, J.-Y., &amp; Schmidt, L. (2024). Growth mindset is associated with mastery goals in adulthood. Europe’s Journal of Psychology, 20(3), 191–201. https://doi.org/10.5964/ejop.11915

Dweck, C. S., &amp; Leggett, E. L. (1988). A social-cognitive approach to motivation and personality. Psychological Review, 95(2), 256–273. https://doi.org/10.1037/0033-295X.95.2.256

Gunderson, E. A., Sorhagen, N. S., Gripshover, S. J., Dweck, C. S., Goldin-Meadow, S., &amp; Levine, S. C. (2018). Parent praise to toddlers predicts fourth grade academic achievement via children’s incremental mindsets. Developmental Psychology, 54(3), 397–409. https://doi.org/10.1037/dev0000444

Macnamara, B. N., &amp; Burgoyne, A. P. (2023). Do growth mindset interventions impact students’ academic achievement? A systematic review and meta-analysis with recommendations for best practices. Psychological Bulletin, 149(3–4), 133–173. https://doi.org/10.1037/bul0000352

Mesler, R. M., Corbin, C. M., &amp; Martin, B. H. (2021). Teacher mindset is associated with development of students’ growth mindset. Journal of Applied Developmental Psychology, 76, 101299. https://doi.org/10.1016/j.appdev.2021.101299

Mueller, C. M., &amp; Dweck, C. S. (1998). Praise for intelligence can undermine children’s motivation and performance. Journal of Personality and Social Psychology, 75(1), 33–52. https://doi.org/10.1037/0022-3514.75.1.33

Sarrasin, J. B., Nenciovici, L., Foisy, L.-M. B., Allaire-Duquette, G., Riopel, M., &amp; Masson, S. (2018). Effects of teaching the concept of neuroplasticity to induce a growth mindset on motivation, achievement, and brain activity: A meta-analysis. Trends in Neuroscience and Education, 12, 22–31. https://doi.org/10.1016/j.tine.2018.07.003

Seligman, M. E., &amp; Maier, S. F. (1967). Failure to escape traumatic shock. Journal of Experimental Psychology, 74(1), 1–9. https://doi.org/10.1037/h0024514

Yu, J., &amp; McLellan, R. (2020). Same mindset, different goals and motivational frameworks: Profiles of mindset-based meaning systems. Contemporary Educational Psychology, 62, 101901. https://doi.org/10.1016/j.cedpsych.2020.101901

}}


==External links==
* [https://www.youtube.com/watch?v=hiiEeMN7vbQ Developing a Growth Mindset with Carol Dweck] (Youtube.com)
* [https://www.youtube.com/watch?v=J-swZaKN2Ic The Power of Yet | Carol S Dweck | Tedx] (Youtube.com)

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      <text bytes="980" sha1="ezpmasmhsprlywymcaoypap1f912re0" xml:space="preserve">== Personal Description ==
My Name is ''Clare Kopplemann'' and I am studying [[Motivation and emotion|Motivation and Emotion]] at the [https://www.canberra.edu.au/ University of Canberra] as a part of my bachelor of Science in Psychology. I am a third year student with a passion for understanding people and behaviour along with [[w:Physiological_psychology|physiological psychology.]]

My personal interests and hobbies include 

* AFL &amp; AFLW (Supporting the Carlton Football Club) 
* Travelling 
* Cooking and Baking 
* Bar and Gaming work 
* Caring for my domestic and stray pet cats 

== Linked-In ==

* [https://www.linkedin.com/in/clare-kopplemann-5a3a972a0/ linked-in.com-clare-kopplemann]

== The Book Chapter I am Currently Working On ==

* [[Motivation and emotion/Book/2026/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine]]

== Social Contribution ==

# [[Motivation and emotion/Book/2026/Developing a growth mindset|Restructured Reference List]]</text>
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      <comment>/* Social Contribution */</comment>
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      <text bytes="1083" sha1="8wa95hhlhzzls3odtwsg36am2ygk0xs" xml:space="preserve">== Personal Description ==
My Name is ''Clare Kopplemann'' and I am studying [[Motivation and emotion|Motivation and Emotion]] at the [https://www.canberra.edu.au/ University of Canberra] as a part of my bachelor of Science in Psychology. I am a third year student with a passion for understanding people and behaviour along with [[w:Physiological_psychology|physiological psychology.]]

My personal interests and hobbies include 

* AFL &amp; AFLW (Supporting the Carlton Football Club) 
* Travelling 
* Cooking and Baking 
* Bar and Gaming work 
* Caring for my domestic and stray pet cats 

== Linked-In ==

* [https://www.linkedin.com/in/clare-kopplemann-5a3a972a0/ linked-in.com-clare-kopplemann]

== The Book Chapter I am Currently Working On ==

* [[Motivation and emotion/Book/2026/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine]]

== Social Contribution ==

# [[Motivation and emotion/Book/2026/Developing a growth mindset|Restructured Reference List]]
# [[Motivation and emotion/Book/2026/Developing a growth mindset|Provided feedback and encouragement]]</text>
      <sha1>8wa95hhlhzzls3odtwsg36am2ygk0xs</sha1>
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    <revision>
      <id>2830671</id>
      <parentid>2830668</parentid>
      <timestamp>2026-09-03T04:42:02Z</timestamp>
      <contributor>
        <username>Ckopplemann</username>
        <id>3108346</id>
      </contributor>
      <comment>/* Social Contribution */</comment>
      <origin>2830671</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1248" sha1="glvu3s18bwdkzwieo7p66t2ekks7zzw" xml:space="preserve">== Personal Description ==
My Name is ''Clare Kopplemann'' and I am studying [[Motivation and emotion|Motivation and Emotion]] at the [https://www.canberra.edu.au/ University of Canberra] as a part of my bachelor of Science in Psychology. I am a third year student with a passion for understanding people and behaviour along with [[w:Physiological_psychology|physiological psychology.]]

My personal interests and hobbies include 

* AFL &amp; AFLW (Supporting the Carlton Football Club) 
* Travelling 
* Cooking and Baking 
* Bar and Gaming work 
* Caring for my domestic and stray pet cats 

== Linked-In ==

* [https://www.linkedin.com/in/clare-kopplemann-5a3a972a0/ linked-in.com-clare-kopplemann]

== The Book Chapter I am Currently Working On ==

* [[Motivation and emotion/Book/2026/Pleasure anticipation and dopamine|Pleasure anticipation and dopamine]]

== Social Contribution ==

# [[Motivation and emotion/Book/2026/Developing a growth mindset|Restructured Reference List]]
# [[Motivation and emotion/Book/2026/Developing a growth mindset|Provided feedback and encouragement]]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345?entry_id=813682 Added a reply to UClearns discussion page ; How do you motivate yourself?]</text>
      <sha1>glvu3s18bwdkzwieo7p66t2ekks7zzw</sha1>
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  <page>
    <title>User:U3254168</title>
    <ns>2</ns>
    <id>331308</id>
    <revision>
      <id>2830621</id>
      <parentid>2823594</parentid>
      <timestamp>2026-09-03T00:37:46Z</timestamp>
      <contributor>
        <username>U3254168</username>
        <id>3106320</id>
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      <origin>2830621</origin>
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      <format>text/x-wiki</format>
      <text bytes="1057" sha1="b6ht2vptg80cntfqxynfph6m639tpax" xml:space="preserve">== About me: ==
[[File:Relaxed female domestic cat.jpg|thumb|'''Figure 1.''' Cats are adorable companions.]]
My name is Zoe Terrill. I am a third year student studying a Bachelor of Laws with a breadth major in Psychology at the [https://www.canberra.edu.au/ University of Canberra]. I am currently undertaking the course [[motivation and emotion]].

Some of my hobbies include:

* Spending time with my cats
* [[w:Crochet|Crocheting]]
* Reading

== Book chapter I'm working on ==
[[Motivation and emotion/Book/2026/Empathy and jury decision-making|Empathy and jury decision-making]]

== Social contributions ==

# [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FFuture_orientation_and_criminal_behaviour&amp;diff=2823593&amp;oldid=2767961 Edit someone's page]
# [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FGetting_started&amp;diff=2827811&amp;oldid=2823598 Comment on someone's page]
# [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 Post on the discussion board]</text>
      <sha1>b6ht2vptg80cntfqxynfph6m639tpax</sha1>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Irritability</title>
    <ns>1</ns>
    <id>331319</id>
    <revision>
      <id>2830636</id>
      <parentid>2830076</parentid>
      <timestamp>2026-09-03T02:30:54Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
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      <text bytes="5195" sha1="rqowbiz0w24uej1uuoijid90j7zzh4t" xml:space="preserve">== Reference material ==

Hi, this 2026 reference may be helpful for your chapter.

Anna Stumps, Leah D. Church, Nadia Bounoua, Lea R. Dougherty, Jeffrey M. Spielberg, The relationship between youth irritability and neural circuitry related to emotion regulation, Journal of Affective Disorders, Volume 396, 2026,120846,ISSN 0165-0327,&lt;nowiki&gt;https://doi.org/10.1016/j.jad.2025.120846&lt;/nowiki&gt;.https://www.sciencedirect.com/science/article/pii/S0165032725022888 [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 01:48, 20 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3275992|Hi [[User:U3275992|U3275992]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 1 September 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# Remove colons from the ends of sub-headings
&lt;!-- Alignment with focus questions --&gt;
# Develop closer alignment between sub-title, focus questions, and top-level headings
|3=
&lt;!-- Overview--&gt;
# Very good
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A promising description of the problem/topic is planned or presented
# Keep this description simple and user-friendly
&lt;!-- Style --&gt;
# Use present, rather than future, tense (fixed)
&lt;!-- Focus questions --&gt;
# Use open- rather then close-ended focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Koints are well developed
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
&lt;!-- Writing style --&gt;
# The writing style is easy to follow
# Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective
&lt;!-- Conclusion --&gt;
# Conclusion hasn't been developed
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented and captioned
&lt;!-- Caption --&gt;
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Place each quiz question in the most relevant section
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
&lt;!-- Systematic reviews --&gt;
# At least one relevant systematic review and/or meta-analysis has been identified
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Basic
## One of two link types provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Excellent
|9=
&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 1 September 2026 (UTC)</text>
      <sha1>rqowbiz0w24uej1uuoijid90j7zzh4t</sha1>
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  <page>
    <title>User:Zozo06</title>
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      <timestamp>2026-09-03T09:41:11Z</timestamp>
      <contributor>
        <username>~2026-47007-35</username>
        <id>3110217</id>
      </contributor>
      <comment>added self blurb in &quot;about me&quot;</comment>
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      <text bytes="1267" sha1="6yv5fxdgdhlu8glqmmqdfaumrfd9d4u" xml:space="preserve">== About me ==
Hi my name is Zoe (Zo), 

I work in Learning and Development (L&amp;D) within an HR team in the Australian Public Service, and I am currently in my third year, studying a Bachelor of Science in Psychology at the University of Canberra (UC) alongside working. I'm genuinely passionate about the intersection of HR and psychology, much of the theory I've learnt whilst completing this degree feels directly transferable to my current role and to HR more broadly. My long-term career goal is to hopefully become an [[w:Industrial and organizational psychology|industrial psychologist]]. or a HR specialist. 

[https://uclearn.canberra.edu.au/courses/20143/assignments/syllabus motivation and emotion]

Within psychology, I have a particular interest in neurodiversity and how diverse considerations can be meaningfully applied across the field, as well as [[w:Emic and etic|emic and etic]] perspectives in understanding behaviour across cultural contexts.

Outside of my academic pursuits, I very much adore most animals and have a sweet spot for horses. As I have been horse riding nearly 10 years.

- template is on topic page. 

== Hobbies ==

# [[w:Equestrianism|horse riding]]
# eating and playing sims 4 


== Book chapter ==

== Social contributions ==</text>
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      <id>2830729</id>
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      <timestamp>2026-09-03T09:57:22Z</timestamp>
      <contributor>
        <username>Zozo06</username>
        <id>3108967</id>
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      <comment>Added links to &quot;about me&quot; and added in my specific book chapter</comment>
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      <text bytes="1678" sha1="kseu4lapqctpl6fnljd6izam749wajc" xml:space="preserve">== About me ==
Hiya, 

My name is Zoe or Zo for short, and I work in the [https://www.aihr.com/blog/learning-and-development/ Learning and Development (L&amp;D) team] within my department in the Australian Public Service. I am currently in my third year, studying a Bachelor of Science in Psychology at the [https://www.canberra.edu.au/ University of Canberra] (UC) alongside working.  

I'm genuinely passionate about the intersection of Human Resource Management (HR) and psychology, much of the theory I've learnt whilst completing this degree feels directly transferable to my current role and to HR more broadly. My long-term career goal is to hopefully become an [[w:Industrial and organizational psychology|industrial psychologist]]. or a HR specialist. 

Within my studies, I have a particular interest in neurodiversity and how diverse considerations can be meaningfully applied across the field, as well as [[w:Emic and etic|emic and etic]] perspectives in understanding behaviour across cultural contexts.

And outside of my academic pursuits, I very much adore most animals and have a sweet spot for horses. As I have been horse riding nearly 10 years.
== Book chapter ==
The topic I have chosen for my book chapter is 

[[Motivation and emotion/Book/2026/Competence motivation in self-determination theory|Competence motivation in self-determination theory]]

I think this topic relates to almost everyone in some way, and it's more layered than it first appears particularly in how it connects back to diversity and individual differences. I'm keen to learn more about it and excited to share my findings with anyone who reads this chapter.

== Social contributions ==</text>
      <sha1>kseu4lapqctpl6fnljd6izam749wajc</sha1>
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  <page>
    <title>User:P U3270518</title>
    <ns>2</ns>
    <id>331388</id>
    <revision>
      <id>2830733</id>
      <parentid>2830366</parentid>
      <timestamp>2026-09-03T11:17:01Z</timestamp>
      <contributor>
        <username>P U3270518</username>
        <id>3106535</id>
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      <comment>/* Social contributions */</comment>
      <origin>2830733</origin>
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      <text bytes="4501" sha1="gaa0npbq9o7y5t7cw2x9c0kikluigqa" xml:space="preserve">== About me ==
Hello Everyone,

I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&amp;rls=en&amp;q=University+of+canberra&amp;ie=UTF-8&amp;oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit.

My Linkedln profile:  https://www.linkedin.com/in/palak-kathiriya-6b7048289/

== Hobbies ==

* Dancing
* Hiking
* Travelling to different countries 
* Painting 
* Cycling 
* Nature exploration
** Birdwatching
** Gardening 

== Book Chapter I'm working on ==
I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? 

Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] 

== Social contributions ==

# 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&amp;diff=2826658&amp;oldid=2826657 Added the template and the title for this page - Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)]
# 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&amp;diff=2826589&amp;oldid=2826579 Fixed spelling error in one of the focus question - Perfectionism and procrastination What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)]
# 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]]
# 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)]
#8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on UC Learn about how to use GenAI in the unit and what are the expectations if we use GenAI]
# 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&amp;diff=2830323&amp;oldid=2761410 Fixed spelling error in book chapter - Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)]
# 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&amp;diff=prev&amp;oldid=2830325 Changed few sentences to make it grammatically better - Grit and academic achievement  What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)]
# 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830328&amp;oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)]
# 10: 54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830330&amp;oldid=2830328 Added reference to support one claim in overview section - Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)]
#12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)]
#3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&amp;diff=2830732&amp;oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow -Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)]
#3 September 2026:
#3 September 2026:</text>
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    <revision>
      <id>2830734</id>
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      <timestamp>2026-09-03T11:17:23Z</timestamp>
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        <id>3106535</id>
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      <text bytes="4510" sha1="7g3e6v6nx4nu82tjsvwwid4r3f4jjzq" xml:space="preserve">== About me ==
Hello Everyone,

I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&amp;rls=en&amp;q=University+of+canberra&amp;ie=UTF-8&amp;oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit.

My Linkedln profile:  https://www.linkedin.com/in/palak-kathiriya-6b7048289/

== Hobbies ==

* Dancing
* Hiking
* Travelling to different countries 
* Painting 
* Cycling 
* Nature exploration
** Birdwatching
** Gardening 

== Book Chapter I'm working on ==
I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? 

Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] 

== Social contributions ==

# 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&amp;diff=2826658&amp;oldid=2826657 Added the template and the title for this page - Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)]
# 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&amp;diff=2826589&amp;oldid=2826579 Fixed spelling error in one of the focus question - Perfectionism and procrastination What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)]
# 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]]
# 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)]
#8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on UC Learn about how to use GenAI in the unit and what are the expectations if we use GenAI]
# 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&amp;diff=2830323&amp;oldid=2761410 Fixed spelling error in book chapter - Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)]
# 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&amp;diff=prev&amp;oldid=2830325 Changed few sentences to make it grammatically better - Grit and academic achievement  What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)]
# 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830328&amp;oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)]
# 10: 54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830330&amp;oldid=2830328 Added reference to support one claim in overview section - Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)]
#12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)]
#9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&amp;diff=2830732&amp;oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow -Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)]
#3 September 2026:
#3 September 2026:</text>
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    <revision>
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      <timestamp>2026-09-03T11:31:38Z</timestamp>
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        <username>P U3270518</username>
        <id>3106535</id>
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      <text bytes="4957" sha1="n1ssnxb6gsua7s5vxeicvo30i3p55m6" xml:space="preserve">== About me ==
Hello Everyone,

I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&amp;rls=en&amp;q=University+of+canberra&amp;ie=UTF-8&amp;oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit.

My Linkedln profile:  https://www.linkedin.com/in/palak-kathiriya-6b7048289/

== Hobbies ==

* Dancing
* Hiking
* Travelling to different countries 
* Painting 
* Cycling 
* Nature exploration
** Birdwatching
** Gardening 

== Book Chapter I'm working on ==
I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? 

Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] 

== Social contributions ==

# 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&amp;diff=2826658&amp;oldid=2826657 Added the template and the title for this page - Title: Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)]
# 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&amp;diff=2826589&amp;oldid=2826579 Fixed spelling error in one of the focus question - Title:  Perfectionism and procrastination - What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)]
# 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - Title: Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]]
# 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)]
#8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on UC Learn about how to use GenAI in the unit and what are the expectations if we use GenAI]
# 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&amp;diff=2830323&amp;oldid=2761410 Fixed spelling error in book chapter - Title: Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)]
# 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&amp;diff=prev&amp;oldid=2830325 Changed few sentences to make it grammatically better - Title: Grit and academic achievement  What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)]
# 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830328&amp;oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - Title: Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)]
# 10: 54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830330&amp;oldid=2830328 Added reference to support one claim in overview section - Title: Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)]
#12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)]
#9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&amp;diff=2830732&amp;oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow - Title: Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)]
#9:27 pm, 3 September 2026: [[Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being|Provided suggestion for wikiversity book chapter page and clarified that the user page should be separate than the book chapter - Title: Outdoor play and children's emotional well-being - How does outdoor play influence children's emotional well-being? Overview - (Book Chapter, 2025)]]
#3 September 2026:</text>
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      <text bytes="4995" sha1="j73uiqyy8ksys8yfadwmxuw4zg7y3hc" xml:space="preserve">== About me ==
Hello Everyone,

I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&amp;rls=en&amp;q=University+of+canberra&amp;ie=UTF-8&amp;oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit.

My Linkedln profile:  https://www.linkedin.com/in/palak-kathiriya-6b7048289/

== Hobbies ==

* Dancing
* Hiking
* Travelling to different countries 
* Painting 
* Cycling 
* Nature exploration
** Birdwatching
** Gardening 

== Book Chapter I'm working on ==
I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? 

Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] 

== Social contributions ==

# 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&amp;diff=2826658&amp;oldid=2826657 Added the template and the title for this page - Title: Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)]
# 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&amp;diff=2826589&amp;oldid=2826579 Fixed spelling error in one of the focus question - Title:  Perfectionism and procrastination - What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)]
# 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - Title: Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]]
# 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)]
#8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on UC Learn about how to use GenAI in the unit and what are the expectations if we use GenAI]
# 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&amp;diff=2830323&amp;oldid=2761410 Fixed spelling error in book chapter - Title: Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)]
# 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&amp;diff=prev&amp;oldid=2830325 Changed few sentences to make it grammatically better - Title: Grit and academic achievement  What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)]
# 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830328&amp;oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - Title: Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)]
# 10: 54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&amp;diff=2830330&amp;oldid=2830328 Added reference to support one claim in overview section - Title: Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)]
#12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)]
#9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&amp;diff=2830732&amp;oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow - Title: Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)]
#9:27 pm, 3 September 2026: [[Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being|Provided suggestion for wikiversity book chapter page and clarified that the user page should be separate than the book chapter - Title: Outdoor play and children's emotional well-being - How does outdoor play influence children's emotional well-being? Overview - (Book Chapter, 2025)]]
#4 September 2026:
#5 September 2026:
#6 September 2026:</text>
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    <title>Motivation and emotion/Book/2026/Creative inspiration and effort</title>
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      <text bytes="10625" sha1="6997nl97l09jyxt3gzdqar60zmv6k6l" xml:space="preserve">{{title|Creative inspiration and effort:&lt;br&gt;How do inspiration and effort interact during the creative process?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=15}}
Imagine you have been staring at a blank page for hours, trying to come up with an original idea for a creative project. Just when you decide to take a break, an idea suddenly comes to you. You feel excited and inspired, and you immediately start working. At first, everything seems to flow naturally and you feel confident about what you are creating.

As you continue, however, the excitement begins to fade. The idea is harder to develop than you expected, and finishing the project requires concentration, persistence, and several changes along the way. Even without the same initial feeling of inspiration, you keep working and experimenting with different possibilities.

While doing this, you unexpectedly discover new ideas that make the project even better than you first imagined. This raises an interesting question: Does inspiration drive us to put effort into creative work, or can the effort we put into creating also lead to new inspiration?
|}
[[File:Artist painting at home. Colorful ballerina.jpg|thumb|'''Figure 1'''. An artist transforming creative inspiration into action through sustained effort.]]
* '''Creative inspiration can give people new ideas and motivate them to start creating.''' It can also help people see new possibilities and feel more interested in developing an idea.
* '''Inspiration alone may not be enough to produce a creative outcome.''' Turning an initial idea into something meaningful often requires effort, persistence, problem-solving, and making changes along the way.
* '''Inspiration and effort may work together during the creative process.''' Inspiration can encourage someone to put more effort into their work, while continuing to work on an idea may also lead to new ideas and further inspiration.
* '''Understanding the relationship between inspiration and effort can help explain how creativity develops.''' It can show how an initial moment of inspiration may eventually become a completed creative outcome through continued effort.
{{RoundBoxTop|theme=15}}
'''Focus Questions''' 
* What is creative inspiration and what role does it play in the creative process?
* What role does effort play in developing creative ideas?
* How does inspiration influence effort during the creative process?
* How can effort lead to further inspiration during the creative process?
* How do inspiration and effort work together to produce creative outcomes?
|}

== Creative inspiration ==
* Creative inspiration is a motivational state that encourages people to turn ideas into action. It can motivate people to develop or express new possibilities (Oleynick et al., 2014).
* Inspiration involves three main characteristics: evocation, transcendence, and approach motivation. Together, these describe how inspiration is triggered, creates new possibilities, and motivates action (Thrash &amp; Elliot, 2003).
* Inspiration can involve being “inspired by” something and then “inspired to” act. This can help turn an inspiring idea or experience into creative activity (Oleynick et al., 2014).
* Creative ideas and inspiration are related but different. Creative ideas can come first and then produce inspiration that motivates people to develop those ideas (Thrash et al., 2010).

* Inspiration can help transform initial ideas into creative outcomes. Research found that inspiration helped carry the creativity of an initial idea into the finished product (Thrash et al., 2010).

== Effort in the creative process ==

* Effort helps develop and improve creative ideas. Turning an initial idea into a finished product often requires persistence, problem-solving, revision, and continued work (Lucas &amp; Nordgren, 2015).
* Inspiration and effort can contribute differently to creative work. Inspiration predicted creativity, while effort predicted the technical quality of creative products (Thrash et al., 2010).
* Effort can support the refinement of creative ideas. Creative work often involves deliberate thinking, making changes, and revising ideas as they develop (Oleynick et al., 2014).
* Persistence can improve creative performance. People often underestimate the value of continuing with a creative task, even though persistence can lead to more creative ideas (Lucas &amp; Nordgren, 2015).

* Effort and inspiration are not necessarily opposites. They can occur together and contribute in different ways throughout the creative process (Oleynick et al., 2014).

== From inspiration to effort ==

* Inspiration can motivate people to act on creative ideas. Feeling inspired can encourage a person to move beyond having an idea and begin developing it (Oleynick et al., 2014). 
* Inspiration and effort are not opposites. Research suggests that people can experience inspiration while also putting substantial effort into their creative work (Thrash et al., 2010).
* Higher inspiration can be associated with greater effort. Writers who felt more inspired also tended to report putting more effort into their writing (Thrash et al., 2010).
* Inspiration may help connect an initial idea with the motivation needed to develop it. The process can move from being “inspired by” an idea to being “inspired to” act on it (Oleynick et al., 2014).

* Inspiration and effort can therefore complement each other during creativity. Research suggests that both can contribute to creative work, although they may contribute in different ways (Oleynick et al., 2014).

== Interaction between inspiration and effort ==

* Inspiration and effort can work together during creative work. Research shows a positive relationship between inspiration and effort rather than the two being opposites (Thrash et al., 2010).
* Inspiration and effort can contribute in different ways. Inspiration has been associated with greater creativity, while effort has been linked more strongly with the technical quality of creative work (Thrash et al., 2010).
* Inspiration can help generate and express creative possibilities, while effort helps develop and refine them. Both can therefore contribute to the creative process in different ways (Oleynick et al., 2014).
* Creative work can involve both spontaneous and deliberate processes. Inspiration may provide motivation and direction, while sustained effort helps turn ideas into completed creative work (Oleynick et al., 2014). 

== Supporting inspiration and sustained creative effort ==

* [[Motivation and emotion/Book/2024/Intrinsic motivation and creativity|Intrinsic motivation]] is positively related to creativity. People with greater intrinsic motivation tend to produce more creative outcomes (Jesus et al., 2013).
* A supportive environment can help creative motivation. Having autonomy, encouragement, and opportunities to explore ideas can make it easier for people to engage in creative work (Amabile &amp; Pratt, 2016).
* Persistence can help when inspiration begins to fade. Continuing to work on a difficult creative task can lead to more creative ideas than people often expect (Lucas &amp; Nordgren, 2015).
* Taking breaks may also support creative thinking. Research suggests that stepping away from a problem for a period of time can sometimes improve later creative performance (Sio &amp; Ormerod, 2009). 

* Supporting creativity may therefore involve both encouraging inspiration and maintaining effort. Creative outcomes can benefit from conditions that support motivation, persistence, and continued development of ideas (Amabile &amp; Pratt, 2016).

== Conclusion ==

* Inspiration and effort both contribute to creativity, but they play different roles throughout the creative process.
* Inspiration can provide ideas and motivation, while effort helps develop, improve, and complete those ideas.
* Effort can keep the creative process moving when inspiration fades, through persistence, revision, and problem-solving.
* Overall, inspiration and effort work together, helping transform initial ideas into developed creative outcomes.

== See also ==
[[w:Creativity|Creativity]] (Wikipedia) 

[[w:Motivation|Motivation]] (Wikipedia)  

[[Motivation and emotion/Book/2014/Creativity and emotion|Creativity and emotion]] (Book Chapter, 2014)

[[Motivation and emotion/Book/2016/Artistic creation motivation|Artistic creation motivation]] (Book Chapter,2016)  

== References ==
{{Hanging indent|1=
Amabile, T. M., &amp; Pratt, M. G. (2016). The dynamic componential model of creativity and innovation in organizations: Making progress, making meaning. Research in Organizational Behavior, 36, 157–183. https://doi.org/10.1016/j.riob.2016.10.001

Jesus, S. N., Rus, C. L., Lens, W., &amp; Imaginário, S. (2013). Intrinsic motivation and creativity related to product: A meta-analysis of the studies published between 1990–2010. Creativity Research Journal, 25(1), 80–84. https://doi.org/10.1080/10400419.2013.752235

Lucas, B. J., &amp; Nordgren, L. F. (2015). People underestimate the value of persistence for creative performance. Journal of Personality and Social Psychology, 109(2), 232–243. https://doi.org/10.1037/pspa0000030

Oleynick, V. C., Thrash, T. M., LeFew, M. C., Moldovan, E. G., &amp; Kieffaber, P. D. (2014). The scientific study of inspiration in the creative process: Challenges and opportunities. Frontiers in Human Neuroscience, 8, Article 436. https://doi.org/10.3389/fnhum.2014.00436

Sio, U. N., &amp; Ormerod, T. C. (2009). Does incubation enhance problem solving? A meta-analytic review. Psychological Bulletin, 135(1), 94–120. https://doi.org/10.1037/a0014212

Thrash, T. M., &amp; Elliot, A. J. (2003). Inspiration as a psychological construct. Journal of Personality and Social Psychology, 84(4), 871–889. https://doi.org/10.1037/0022-3514.84.4.871

Thrash, T. M., Maruskin, L. A., Cassidy, S. E., Fryer, J. W., &amp; Ryan, R. M. (2010). Mediating between the muse and the masses: Inspiration and the actualization of creative ideas. Journal of Personality and Social Psychology, 98(3), 469–487. https://doi.org/10.1037/a0017907
}}

== External Links ==
[https://www.youtube.com/watch?v=lSaNEcNprzw The Art of Creative Inspiration]  (Youtube video)

[https://www.apa.org/monitor/2022/04/cover-science-creativity The Science behind creativity] ( American Psychological Association) 

[https://www.youtube.com/watch?v=AzJcNwDeS2c Inspiration: Find it, Use it, Share it] ( Youtube video)

[https://www.psychologytoday.com/au/blog/explorations-of-the-mind/202309/creativity-and-inspiration-thinking-outside-the-box Creativity and inspiration]  (Psychology today)</text>
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    <title>Talk:Motivation and emotion/Book/2026/Self-disclosure and emotional intimacy</title>
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      <text bytes="6408" sha1="ozixlquhnigyf8xp08br9vz1z22osdm" xml:space="preserve">== Scenario Suggestion - From U3275775 ==

For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC)

== Resource ==

Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers!

https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC)

== Extra Resource ==

Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) 

Reis, H. T., &amp; Shaver, P. (1988). &lt;nowiki&gt;''&lt;/nowiki&gt;Intimacy as an Interpersonal Process&lt;nowiki&gt;''&lt;/nowiki&gt; (pp. 367–389). Routledge. &lt;nowiki&gt;https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process&lt;/nowiki&gt; [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:28, 31 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
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&lt;!-- Title --&gt;
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&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
&lt;!-- Alignment with focus questions --&gt;
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
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&lt;!-- Description --&gt;
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# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
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# Highlight the most relevant theories and synthesise the best research on the topic
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&lt;!-- Cite --&gt;
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&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
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# Promising use of quiz question(s)
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&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
# Only cite sources you have consulted
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## use dois where available instead of other links
## include hyperlinked dois
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&lt;!-- Resources --&gt;
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&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Brief description about self – consider expanding
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
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&lt;!-- Social contribution --&gt;
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:28, 31 August 2026 (UTC)</text>
      <sha1>ozixlquhnigyf8xp08br9vz1z22osdm</sha1>
    </revision>
    <revision>
      <id>2830700</id>
      <parentid>2830696</parentid>
      <timestamp>2026-09-03T06:49:52Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>- basic development of bullet-points with some citations for the proposed sections</comment>
      <origin>2830700</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6491" sha1="5jh1cpo958h5io9o4yu7iww083mulk5" xml:space="preserve">== Scenario Suggestion - From U3275775 ==

For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC)

== Resource ==

Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers!

https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC)

== Extra Resource ==

Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) 

Reis, H. T., &amp; Shaver, P. (1988). &lt;nowiki&gt;''&lt;/nowiki&gt;Intimacy as an Interpersonal Process&lt;nowiki&gt;''&lt;/nowiki&gt; (pp. 367–389). Routledge. &lt;nowiki&gt;https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process&lt;/nowiki&gt; [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:28, 31 August 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
&lt;!-- Alignment with focus questions --&gt;
# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
|3=
&lt;!-- Overview--&gt;
# Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Description --&gt;
# A promising description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
# Reasonably good alignment between focus questions and heading structure, but consider closer alignment
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Promising development - basic development of bullet-points with some citations for the proposed sections
# Highlight the most relevant theories and synthesise the best research on the topic
# Provide more detailed edit summaries
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
&lt;!-- Conclusion --&gt;
# Conclusion is well underway
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented and captioned
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
&lt;!-- Size --&gt;
# Consider increasing image size(s) (especially if they have text) to make them easier to view
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
# Only cite sources you have consulted
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## use dois where available instead of other links
## include hyperlinked dois
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
&lt;!-- External links --&gt;
# External links
## Excellent
|9=
&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Brief description about self – consider expanding
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
#* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]]
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:28, 31 August 2026 (UTC)</text>
      <sha1>5jh1cpo958h5io9o4yu7iww083mulk5</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Self-concept and motivation</title>
    <ns>1</ns>
    <id>331472</id>
    <revision>
      <id>2830685</id>
      <parentid>2829228</parentid>
      <timestamp>2026-09-03T06:29:22Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Removed &quot;# There are two &quot;The influence of motivation on self-concept&quot; headings&quot;</comment>
      <origin>2830685</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4576" sha1="0lb2lmmj1jxjudgfj5c0gkne7e0hik6" xml:space="preserve">== Feedback on chapter clarity  ==

Hi, I really liked how you have used Ash’s swimming scenario throughout the chapter to connect self-concept and motivation to a clear example. The use of psychological theories and research also provides support for the key ideas. One suggestion would be to simplify some of the more detailed sections where several theories and studies are discussed together. This could make the main points clearer and easier to follow and understand.
Well done you have done an amazing job altogether!! [[User:U3253354|U3253354]] ([[User talk:U3253354|discuss]] • [[Special:Contributions/U3253354|contribs]]) 02:36, 25 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# There may be too many sub-headings given that the plan already exceeds the maximum word count for a full chapter e.g., consider simplifying the &quot;understanding self-concept&quot; section (i.e., summarise and link to other resources).
# Good to see the intervention section; but this content could be probably be condensed e.g., into some case studies/feature boxes to highlight activities to try
&lt;!-- Alignment with focus questions --&gt;
# Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Promising focus questions
# Develop closer alignment between the sub-title, focus questions, and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# However, it is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Theory and research --&gt;
# Promising balance of theory and research
&lt;!-- Citations --&gt;
# Excellent use of citations
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Promising use of scenarios/examples/case studies
&lt;!-- Tables --&gt;
# Promising use of table(s); I suggest using a table to summarise the relation between SC and M (i.e., directly address the topic) (rather than on general SC info)
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Very good
## Use [[w:Letter case#Sentence casing|sentence casing]]
&lt;!-- User page --&gt;
# Excellent
&lt;!-- Description about self --&gt;
# Excellent description about self provided
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:38, 28 August 2026 (UTC)</text>
      <sha1>0lb2lmmj1jxjudgfj5c0gkne7e0hik6</sha1>
    </revision>
    <revision>
      <id>2830686</id>
      <parentid>2830685</parentid>
      <timestamp>2026-09-03T06:32:44Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Basic -&gt; good alignment between sub-title, focus questions, and headings</comment>
      <origin>2830686</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4634" sha1="oq8x1po1inpk8zbas3oz9jevpef8zhk" xml:space="preserve">== Feedback on chapter clarity  ==

Hi, I really liked how you have used Ash’s swimming scenario throughout the chapter to connect self-concept and motivation to a clear example. The use of psychological theories and research also provides support for the key ideas. One suggestion would be to simplify some of the more detailed sections where several theories and studies are discussed together. This could make the main points clearer and easier to follow and understand.
Well done you have done an amazing job altogether!! [[User:U3253354|U3253354]] ([[User talk:U3253354|discuss]] • [[Special:Contributions/U3253354|contribs]]) 02:36, 25 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic.
# There may be too many sub-headings given that the plan already exceeds the maximum word count for a full chapter e.g., consider simplifying the &quot;understanding self-concept&quot; section (i.e., summarise and link to other resources).
# Good to see the intervention section; but this content could be probably be condensed e.g., into some case studies/feature boxes to highlight activities to try
&lt;!-- Alignment with focus questions --&gt;
# Good alignment between between sub-title, focus questions, and top-level headings. Consider closer alignment (e.g., by using the focus questions as headings)
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A clear description of the problem/topic is planned or presented
# Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios)
&lt;!-- Focus questions --&gt;
# Promising focus questions
# Develop closer alignment between the sub-title, focus questions, and top-level headings
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# However, it is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title
&lt;!-- Theory and research --&gt;
# Promising balance of theory and research
&lt;!-- Citations --&gt;
# Excellent use of citations
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
&lt;!-- Creation --&gt;
# Well done on creating and uploading your own image! {{smile}}
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Promising use of scenarios/examples/case studies
&lt;!-- Tables --&gt;
# Promising use of table(s); I suggest using a table to summarise the relation between SC and M (i.e., directly address the topic) (rather than on general SC info)
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Excellent
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Excellent
## Use alphabetical order
&lt;!-- External links --&gt;
# External links
## Very good
## Use [[w:Letter case#Sentence casing|sentence casing]]
&lt;!-- User page --&gt;
# Excellent
&lt;!-- Description about self --&gt;
# Excellent description about self provided
&lt;!-- Links to profile(s) --&gt;
# Link(s) provided to professional profile(s)
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:38, 28 August 2026 (UTC)</text>
      <sha1>oq8x1po1inpk8zbas3oz9jevpef8zhk</sha1>
    </revision>
  </page>
  <page>
    <title>WikiJournal User Group/Administrative officer</title>
    <ns>0</ns>
    <id>331473</id>
    <revision>
      <id>2830573</id>
      <parentid>2826815</parentid>
      <timestamp>2026-09-02T20:06:11Z</timestamp>
      <contributor>
        <username>OhanaUnited</username>
        <id>18921</id>
      </contributor>
      <minor />
      <comment>/* Budget and reimbursement */</comment>
      <origin>2830573</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11162" sha1="cztl36i2fm3xo8sd3zpl4dwylj068zr" xml:space="preserve">&lt;noinclude&gt;{{WikiJ top menu}}
__NOTOC__

'''Administrative officer''' supports the day-to-day organizational and administrative work of WikiJournal, including financial administration, contractor coordination, grant reporting, and other operational tasks. The role is intended to provide continuity for work that can otherwise be difficult to sustain through volunteer effort alone.

The administrative officer primarily reports to and takes direction from the [[WikiJournal User Group/Administrative board|Administrative Board]], but may also carry out administrative tasks requested by the editorial boards of individual WikiJournals when those tasks are within the scope of the journal and within established WikiJournal policies and budget.

{{wjh_h2|Administrative officer}}
&lt;/noinclude&gt;
*Current administrative officer to be listed here when appointed.
&lt;noinclude&gt;

{{wjs_h2|Role details}}

===Main tasks===
The tasks of the administrative officer are primarily organizational, financial, personnel-related and operational. This allows editorial board members and other contributors, who are largely unpaid volunteers, to focus on editorial decisions, peer review, strategy and scholarly contributions.

Such tasks may include:

*'''Administrative coordination'''
**Maintaining an overview of open administrative tasks, deadlines and recurring obligations.
**Following up on action items from Administrative Board meetings and helping ensure that agreed actions are completed.
**Preparing or organizing administrative documents, records, templates and routine correspondence.
**Helping maintain up-to-date administrative information on WikiJournal wiki pages and shared documentation.

*'''Financial administration and grant support'''
**Assisting with grant applications, grant administration, budget tracking and required grant reporting.
**Helping prepare annual and other periodic financial or activity reports for review and approval by the appropriate board members.
**Organizing documentation for expenses, reimbursements, contractor payments, tax filings and other financial records.
**Alerting the Administrative Board to anticipated budget overruns, unspent funds or administrative expenses requiring decisions.

*'''Personnel and contractor administration'''
**Coordinating contracting, onboarding and offboarding of paid contractors, including [[WikiJournal User Group/Technical editors|technical editors]].
**Maintaining contractor documentation and records of agreed roles, rates, work limits and reporting requirements.
**Coordinating routine payment documentation and helping resolve administrative issues affecting contractors.
**Supporting recruitment logistics, including posting openings, organizing applications, scheduling interviews and maintaining records of selection decisions.

*'''Communications and organizational support'''
**Assisting with routine communications involving WikiJournal participants, contractors, partner organizations and funders when delegated.
**Supporting preparation of meeting agendas, administrative updates and follow-up communications.
**Helping maintain institutional continuity by documenting recurring processes and updating administrative instructions.

===Relationship to editorial work===
The administrative officer does not replace the editorial boards, editors-in-chief, handling editors, peer-review coordinators or technical editors.

Publication decisions and other editorial judgments remain with the relevant editorial bodies. The administrative officer may support these processes administratively, for example through tracking, reminders, documentation or coordination, but should not independently make editorial decisions unless separately serving in an authorized editorial role.

===Relationship to technical editors===
The administrative officer may coordinate the contracting, onboarding, documentation and payment administration of [[WikiJournal User Group/Technical editors|technical editors]].

Technical editors continue to perform technical and article-processing work under their own role description. Where useful, existing technical-editor documentation, onboarding materials, contractor templates, task logs and reimbursement procedures may be adapted for use by the administrative officer.

===Recordkeeping and reporting===
The administrative officer should keep a record of paid work performed and how much time was used for each task.

A brief report should normally be presented about monthly to the Administrative Board. This may include:

*Hours worked and major tasks completed.
*Open or delayed tasks requiring board attention.
*Contractor or personnel matters requiring decisions.
*Budget or reimbursement items requiring review.
*Upcoming reporting, filing, renewal or grant deadlines.
*Work performed directly for individual WikiJournal editorial boards.

===Additional tasks===
Depending on the experience of the person selected and the needs of WikiJournal, the administrative officer may also assist with:

*Drafting or updating standard operating procedures and onboarding checklists.
*Maintaining organizational calendars and recurring-deadline lists.
*Supporting grant metrics and impact reporting.
*Coordinating with bookkeeping, tax, banking, insurance, payment or contracting services.
*Helping evaluate lower-cost administrative or contractor-management tools.
*Supporting external communications, partnerships, conference administration or similar organizational projects when assigned.

The ''administrative officer'' should act in accordance with the policies of WikiJournal, including the [[WikiJournal User Group/Bylaws|Bylaws]] and [[WikiJournal User Group/Ethics statement|Ethics statement]], as well as applicable grant and contractual requirements.

Potential conflicts of interest, suspected misconduct, legal concerns or matters outside the officer's authority should be referred to the appropriate board rather than decided independently.

{{wjs_h2|Admin details}}

===Schedule===
The position is intended as a part-time and flexibly scheduled role rather than a fixed daily shift.

The expected workload is approximately '''6 hours per week on average''', although workload may vary depending on grant deadlines, reporting periods, contracting, onboarding and other time-sensitive needs.

*Most work may be performed asynchronously and remotely.
*The administrative officer should normally review WikiJournal administrative communications regularly during the work week.
*Attendance at monthly general WikiJournal meetings should be expected when reasonably possible and counted as paid working time.
*Meetings with the Administrative Board, individual editorial boards, contractors or other WikiJournal participants may also be counted as paid working time when related to the role.
*The Administrative Board may authorize additional or redistributed hours for defined projects, provided funding remains available within the approved budget.

===Budget and reimbursement===
Approximately '''US$8,000''' is allocated to the administrative officer role for the year, while retaining additional grant funding for [[WikiJournal User Group/Technical editors|technical editor]] work and other WikiJournal activities.

The proposed rate is '''US$25 per hour''', with payment based on actual documented hours worked. An annual allocation of US$8,000 provides for up to approximately '''320 hours per year''', corresponding to an average of approximately '''6 hours per week'''.

The Administrative Board may adjust the distribution of hours during the year according to workload and available funding.

Reasonable out-of-pocket expenses incurred specifically for WikiJournal work may be reimbursed when approved in advance by an authorized Administrative Board member or under a standing Board-approved policy. Such expenses should be documented with receipts or equivalent records.

Examples may include:
*Required software or service fees.
*Postage or administrative filing costs.
*Approved travel or other project-specific expenses.

Personal equipment and ordinary home internet or workspace expenses would not normally be reimbursed unless specifically authorized.

===Location and international contracting===
The administrative officer position may be open to applicants internationally, provided that WikiJournal is able to establish a practical and legally appropriate method for contracting and paying the selected applicant in their country of residence.

WikiJournal should confirm the available payment method and any required contractor or tax documentation before paid work begins. Applicants should therefore not necessarily be excluded solely because they reside outside the United States, but appointment may depend on WikiJournal being able to make payments reliably and at reasonable administrative cost.

Where possible, the same hourly rate should apply irrespective of the contractor's country of residence.

===Hiring process===
If or when the WikiJournal User Group has an open administrative officer position, the hiring process may consist of:

*First contacting current or previous [[WikiJournal User Group/Technical editors|technical editors]] and other experienced WikiJournal participants who may already be familiar with WikiJournal workflows and documentation.
*Opening the position to suitable applicants internationally where practical payment and contracting arrangements are available.
*If this does not identify a suitable candidate, advertising the position more broadly within Wikimedia, open-access, academic-publishing and nonprofit communities.
*Requesting a concise application describing relevant administrative, nonprofit, financial, human-resources, contractor-management, Wikimedia or scholarly-publishing experience.
*Confirming that WikiJournal can establish an appropriate contracting and payment arrangement in the shortlisted applicant's country before appointment.
*Verifying the identity of shortlisted applicants before contracting.
*Preferably having approximately 2–3 WikiJournal participants review or interview shortlisted applicants.
*Providing a summary of shortlisted candidates to the Administrative Board.
*Selecting the contractor by consensus of the Administrative Board.

===Desirable experience===
Useful qualifications may include:

*Reliable administrative organization and follow-through.
*Experience with contractor administration, human resources, bookkeeping, grants, nonprofits or project administration.
*Clear written communication and comfort working asynchronously with an international volunteer community.
*Fluency in written and spoken English sufficient for professional communication, meetings, administrative documentation and correspondence.
*Ability to maintain accurate records and appropriately handle confidential administrative information.
*Familiarity with Wikimedia projects, WikiJournal, academic publishing or open-access communities.
*Previous WikiJournal technical-editor or other WikiJournal experience, particularly because existing documentation and workflows may be adapted for this role.

&lt;/noinclude&gt;
[[Category:WikiJournal User Group]]</text>
      <sha1>cztl36i2fm3xo8sd3zpl4dwylj068zr</sha1>
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  <page>
    <title>Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being</title>
    <ns>1</ns>
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    <revision>
      <id>2830735</id>
      <parentid>2825882</parentid>
      <timestamp>2026-09-03T11:26:45Z</timestamp>
      <contributor>
        <username>P U3270518</username>
        <id>3106535</id>
      </contributor>
      <comment>suggestion about book chapter page</comment>
      <origin>2830735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="911" sha1="k1ftgonmaeohsd5tzkuznxmuhgmmg0z" xml:space="preserve">== Overview scenario ==

Hey there!

Super interesting topic you have chosen, as well as an awesome start to hook your readers in. The only additional suggestion I would make is to place your scenario into a feature box, to make it really stand out. Other than that, great start. [[User:LMM26|LMM26]] ([[User talk:LMM26|discuss]] • [[Special:Contributions/LMM26|contribs]]) 13:59, 25 August 2026 (UTC)

== User page == 
 
Hi there, 

Interesting topic.

I have a small suggestion regarding your page. It might be better to create a separate user page for your personal information and social contributions, rather than including it on the same page as your book chapter. This would help keep the book chapter page focused and organised.

Thanks and all the best.--[[User:P U3270518|P U3270518]] ([[User talk:P U3270518|discuss]] • [[Special:Contributions/P U3270518|contribs]]) 11:26, 3 September 2026 (UTC)</text>
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  <page>
    <title>Motivation and emotion/Book/2026/Sun exposure and protection motivation</title>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>added [[Category:Motivation and emotion/Book/Environment]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="10528" sha1="cdmghzru0iisprenjwcodzfc3yjpdht" xml:space="preserve">{{title|Sun exposure and protection motivation&lt;br&gt;What motivates sun exposure and protection behaviours?}}
&lt;div align=center&gt;Edit the wording (and [[w:Stylistic or specialised usage|casing]]) above so that it matches the [[Motivation and emotion/Book/Current|topic list]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not add your name; authorship is shown in the [[Special:History/{{PAGENAME}}|page history]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}} 
[[File:Sunscreen spray.jpg|thumb|right|200px|'''Figure 1.''' Applying sunscreen is a sun-protective behaviour used to reduce harmful UV exposure.]]
Mia is a 22-year-old university student spending a summer afternoon at the beach with friends. She knows that excessive ultraviolet (UV) exposure increases the risk of skin damage and skin cancer, so she brings SPF 50+ sunscreen. However, Mia also likes how she looks with a tan and notices that her friends are intentionally tanning. Although she planned to reapply sunscreen and spend time in the shade, she stays in the sun longer than intended. Mia's behaviour illustrates a common conflict: why might someone who understands the risks of sun exposure still feel motivated to tan, while also wanting to protect their skin?{{RoundBoxBottom}}
* Sun-related behaviour involves competing motivations: people may be motivated to seek sun exposure while simultaneously wanting to protect themselves from harmful ultraviolet (UV) radiation.
* Intentional sun exposure can be motivated by factors such as appearance enhancement, tanning preferences, relaxation, wellbeing and social influences, whereas sun protection is strongly associated with health-preservation motives (Ingledew et al., 2010).
* Knowledge about the risks of UV exposure does not necessarily result in protective behaviour because decisions are also influenced by perceived risk, self-efficacy, social norms and the perceived benefits of tanning.
* Psychological theories, particularly '''protection motivation theory''', can help explain why people adopt or avoid sun-protective behaviours by examining perceptions of threat and people's beliefs about their ability to respond effectively (Rogers, 1975).

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What psychological and social factors motivate intentional sun exposure?
* What motivates people to engage in sun-protection behaviours?
* How does protection motivation theory explain sun-protective behaviour?
* How can psychological research be used to encourage safer sun exposure and protection behaviours?
{{RoundBoxBottom}}
*

==== What motivates intentional sun exposure? ====

* Intentional sun exposure is influenced by multiple psychological motives and cannot be explained solely by a lack of knowledge about the health risks associated with ultraviolet (UV) radiation.
* '''Appearance enhancement''' is an important motivation for sun exposure, as some individuals perceive tanned skin as more attractive and desirable (Leary, 1993).
* Sun exposure may also be motivated by '''wellbeing and relaxation'''. Ingledew et al. (2010) found that appearance-enhancement and wellbeing motives were positively associated with sun-exposure behaviour.

* '''Social influences''' can reinforce tanning behaviour, as attitudes and norms among friends, family and broader culture may shape perceptions of tanning and sun exposure (Hillhouse et al., 2000).

==== What motivates sun-protection behaviour? ====

* Sun-protection behaviours include actions such as using sunscreen, seeking shade, wearing protective clothing and limiting exposure to harmful UV radiation.
* '''Health preservation''' is an important motivation for sun protection. Ingledew et al. (2010) found that health-preservation motives positively influenced sun-protection behaviour.
* Motivation to protect the skin is influenced not only by knowledge of UV risks, but also by whether individuals perceive themselves as vulnerable to harm and believe protective behaviours will be effective (Jackson &amp; Aiken, 2000).

* '''Self-efficacy''' is particularly important because people are more likely to engage in protective behaviours when they feel confident in their ability to perform them successfully (Floyd et al., 2000).

==== How does protection motivation theory explain sun-protective behaviour? ====

* '''Protection motivation theory (PMT)''' proposes that protective behaviour is influenced by two appraisal processes: '''threat appraisal''' and '''coping appraisal''' (Rogers, 1975).
* '''Threat appraisal''' involves evaluating the perceived severity of a threat, such as skin cancer or UV damage, and one's perceived vulnerability to experiencing that threat.
* '''Coping appraisal''' involves evaluating response efficacy, or whether a protective behaviour will work, and self-efficacy, or whether the individual believes they can successfully perform the behaviour.

* Research supports these mechanisms, with greater perceived severity, vulnerability, response efficacy and self-efficacy associated with stronger protective intentions and behaviours (Floyd et al., 2000).

==== How can safer sun behaviour be encouraged? ====

* Effective sun-safety interventions should address both the '''health risks of UV exposure''' and the psychological rewards that can motivate tanning, such as appearance and social approval.
* Protection motivation theory suggests that interventions can encourage protective behaviour by increasing perceived vulnerability and severity while also strengthening '''response efficacy and self-efficacy''' (Floyd et al., 2000).
* Appearance-focused approaches may be useful for young adults. Jones and Leary (1994) found that an appearance-based intervention increased intentions to engage in safer sun behaviour, suggesting that immediate appearance consequences may sometimes be more motivating than distant health risks.

* Sun-safety strategies should therefore combine clear risk information with practical and achievable protective actions, while also challenging social norms that portray tanning as desirable.

==Key points==

* Intentional sun exposure is motivated by factors including appearance enhancement, wellbeing and social influences (Ingledew et al., 2010).
* Sun-protection behaviour is influenced by health-preservation motives, perceived vulnerability and self-efficacy (Ingledew et al., 2010; Jackson &amp; Aiken, 2000).
* Protection motivation theory explains sun-protective behaviour through threat appraisal and coping appraisal, including perceived severity, vulnerability, response efficacy and self-efficacy (Rogers, 1975; Floyd et al., 2000).
* Sun-safety interventions may be more effective when they address both health risks and the appearance and social rewards associated with tanning (Jones &amp; Leary, 1994).

==Figures==

==Learning features==

'''Case study reflection'''

* Revisit Mia's scenario from the Overview. What competing motivations may explain why she continues sun exposure despite knowing the potential health risks?
* Using '''protection motivation theory''', identify how perceived severity, perceived vulnerability, response efficacy, and self-efficacy could influence Mia's sun-protection behaviour.
* What psychological strategy could encourage Mia to use sun protection while also addressing the appearance and social motivations associated with tanning?

==Conclusion==

* Sun exposure and protection behaviours are influenced by competing '''health, appearance, wellbeing and social motivations'''.
* Appearance enhancement and wellbeing can motivate intentional sun exposure, whereas health-preservation motives can encourage protective behaviour.
* '''Protection motivation theory''' helps explain sun-protective behaviour through threat appraisal and coping appraisal, including perceived severity, vulnerability, response efficacy and self-efficacy.
* Effective sun-safety interventions should therefore address both perceptions of health risk and the psychological and social rewards associated with tanning, while strengthening people's confidence in their ability to protect themselves.

== See also ==

* [[w:Health belief model|Health belief model]] (Wikipedia)
* [[w:Protection motivation theory|Protection motivation theory]] (Wikipedia)
* [[w:Sun tanning|Sun tanning]] (Wikipedia)
* [[w:Sunscreen|Sunscreen]] (Wikipedia)
* [[w:Ultraviolet|Ultraviolet radiation]] (Wikipedia)

==References==
{{Hanging indent|1=
Floyd, D. L., Prentice-Dunn, S., &amp; Rogers, R. W. (2000). A meta-analysis of research on protection motivation theory. ''Journal of Applied Social Psychology, 30''(2), 407–429. https://doi.org/10.1111/j.1559-1816.2000.tb02323.x

Hillhouse, J. J., Turrisi, R., &amp; Kastner, M. (2000). Modeling tanning salon behavioral tendencies using appearance motivation, self-monitoring and the theory of planned behavior. ''Health Education Research, 15''(4), 405–414. https://doi.org/10.1093/her/15.4.405

Ingledew, D. K., Ferguson, E., &amp; Markland, D. (2010). Motives and sun-related behaviour. ''Journal of Health Psychology, 15''(1), 8–20. https://doi.org/10.1177/1359105309342292

Jackson, K. M., &amp; Aiken, L. S. (2000). A psychosocial model of sun protection and sunbathing in young women: The impact of health beliefs, attitudes, norms, and self-efficacy for sun protection. ''Health Psychology, 19''(5), 469–478. https://doi.org/10.1037/0278-6133.19.5.469

Jones, J. L., &amp; Leary, M. R. (1994). Effects of appearance-based admonitions against sun exposure on tanning intentions in young adults. ''Health Psychology, 13''(1), 86–90. https://doi.org/10.1037/0278-6133.13.1.86

Leary, M. R., &amp; Jones, J. L. (1993). The social psychology of tanning and sunscreen use: Self-presentational motives as a predictor of health risk. ''Journal of Applied Social Psychology, 23''(17), 1390–1406. https://doi.org/10.1111/j.1559-1816.1993.tb01039.x

Rogers, R. W. (1975). A protection motivation theory of fear appeals and attitude change. ''The Journal of Psychology, 91''(1), 93–114. https://doi.org/10.1080/00223980.1975.9915803
}}

==External links==

* [https://www.cancer.org.au/cancer-information/causes-and-prevention/sun-safety Be SunSmart and sun safety] (Cancer Council Australia)
* [https://www.sunsmart.com.au/ SunSmart] (Cancer Council Victoria)
* [https://www.cancer.org.au/cancer-information/causes-and-prevention/sun-safety/uv-index UV Index] (Cancer Council Australia)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Environment]]</text>
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      <text bytes="10576" sha1="loshn1rezrrljnozsnxpofj0sebyl55" xml:space="preserve">{{title|Sun exposure and protection motivation&lt;br&gt;What motivates sun exposure and protection behaviours?}}
&lt;div align=center&gt;Edit the wording (and [[w:Stylistic or specialised usage|casing]]) above so that it matches the [[Motivation and emotion/Book/Current|topic list]].&lt;br&gt;[[Motivation and emotion/About/Staff|Seek approval]] for any changes.&lt;br&gt;Do not add your name; authorship is shown in the [[Special:History/{{PAGENAME}}|page history]].&lt;/div&gt;
__TOC__

==Overview==
{{RoundBoxTop|theme=3}} 
[[File:Sunscreen spray.jpg|thumb|right|200px|'''Figure 1.''' Applying sunscreen is a sun-protective behaviour used to reduce harmful UV exposure.]]
Mia is a 22-year-old university student spending a summer afternoon at the beach with friends. She knows that excessive ultraviolet (UV) exposure increases the risk of skin damage and skin cancer, so she brings SPF 50+ sunscreen. However, Mia also likes how she looks with a tan and notices that her friends are intentionally tanning. Although she planned to reapply sunscreen and spend time in the shade, she stays in the sun longer than intended. Mia's behaviour illustrates a common conflict: why might someone who understands the risks of sun exposure still feel motivated to tan, while also wanting to protect their skin?{{RoundBoxBottom}}
* Sun-related behaviour involves competing motivations: people may be motivated to seek sun exposure while simultaneously wanting to protect themselves from harmful ultraviolet (UV) radiation.
* Intentional sun exposure can be motivated by factors such as appearance enhancement, tanning preferences, relaxation, wellbeing and social influences, whereas sun protection is strongly associated with health-preservation motives (Ingledew et al., 2010).
* Knowledge about the risks of UV exposure does not necessarily result in protective behaviour because decisions are also influenced by perceived risk, self-efficacy, social norms and the perceived benefits of tanning.
* Psychological theories, particularly '''protection motivation theory''', can help explain why people adopt or avoid sun-protective behaviours by examining perceptions of threat and people's beliefs about their ability to respond effectively (Rogers, 1975).

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What psychological and social factors motivate intentional sun exposure?
* What motivates people to engage in sun-protection behaviours?
* How does protection motivation theory explain sun-protective behaviour?
* How can psychological research be used to encourage safer sun exposure and protection behaviours?
{{RoundBoxBottom}}
*

==== What motivates intentional sun exposure? ====

* Intentional sun exposure is influenced by multiple psychological motives and cannot be explained solely by a lack of knowledge about the health risks associated with ultraviolet (UV) radiation.
* '''Appearance enhancement''' is an important motivation for sun exposure, as some individuals perceive tanned skin as more attractive and desirable (Leary, 1993).
* Sun exposure may also be motivated by '''wellbeing and relaxation'''. Ingledew et al. (2010) found that appearance-enhancement and wellbeing motives were positively associated with sun-exposure behaviour.

* '''Social influences''' can reinforce tanning behaviour, as attitudes and norms among friends, family and broader culture may shape perceptions of tanning and sun exposure (Hillhouse et al., 2000).

==== What motivates sun-protection behaviour? ====

* Sun-protection behaviours include actions such as using sunscreen, seeking shade, wearing protective clothing and limiting exposure to harmful UV radiation.
* '''Health preservation''' is an important motivation for sun protection. Ingledew et al. (2010) found that health-preservation motives positively influenced sun-protection behaviour.
* Motivation to protect the skin is influenced not only by knowledge of UV risks, but also by whether individuals perceive themselves as vulnerable to harm and believe protective behaviours will be effective (Jackson &amp; Aiken, 2000).

* '''Self-efficacy''' is particularly important because people are more likely to engage in protective behaviours when they feel confident in their ability to perform them successfully (Floyd et al., 2000).

==== How does protection motivation theory explain sun-protective behaviour? ====

* '''Protection motivation theory (PMT)''' proposes that protective behaviour is influenced by two appraisal processes: '''threat appraisal''' and '''coping appraisal''' (Rogers, 1975).
* '''Threat appraisal''' involves evaluating the perceived severity of a threat, such as skin cancer or UV damage, and one's perceived vulnerability to experiencing that threat.
* '''Coping appraisal''' involves evaluating response efficacy, or whether a protective behaviour will work, and self-efficacy, or whether the individual believes they can successfully perform the behaviour.

* Research supports these mechanisms, with greater perceived severity, vulnerability, response efficacy and self-efficacy associated with stronger protective intentions and behaviours (Floyd et al., 2000).

==== How can safer sun behaviour be encouraged? ====

* Effective sun-safety interventions should address both the '''health risks of UV exposure''' and the psychological rewards that can motivate tanning, such as appearance and social approval.
* Protection motivation theory suggests that interventions can encourage protective behaviour by increasing perceived vulnerability and severity while also strengthening '''response efficacy and self-efficacy''' (Floyd et al., 2000).
* Appearance-focused approaches may be useful for young adults. Jones and Leary (1994) found that an appearance-based intervention increased intentions to engage in safer sun behaviour, suggesting that immediate appearance consequences may sometimes be more motivating than distant health risks.

* Sun-safety strategies should therefore combine clear risk information with practical and achievable protective actions, while also challenging social norms that portray tanning as desirable.

==Key points==

* Intentional sun exposure is motivated by factors including appearance enhancement, wellbeing and social influences (Ingledew et al., 2010).
* Sun-protection behaviour is influenced by health-preservation motives, perceived vulnerability and self-efficacy (Ingledew et al., 2010; Jackson &amp; Aiken, 2000).
* Protection motivation theory explains sun-protective behaviour through threat appraisal and coping appraisal, including perceived severity, vulnerability, response efficacy and self-efficacy (Rogers, 1975; Floyd et al., 2000).
* Sun-safety interventions may be more effective when they address both health risks and the appearance and social rewards associated with tanning (Jones &amp; Leary, 1994).

==Figures==

==Learning features==

'''Case study reflection'''

* Revisit Mia's scenario from the Overview. What competing motivations may explain why she continues sun exposure despite knowing the potential health risks?
* Using '''protection motivation theory''', identify how perceived severity, perceived vulnerability, response efficacy, and self-efficacy could influence Mia's sun-protection behaviour.
* What psychological strategy could encourage Mia to use sun protection while also addressing the appearance and social motivations associated with tanning?

==Conclusion==

* Sun exposure and protection behaviours are influenced by competing '''health, appearance, wellbeing and social motivations'''.
* Appearance enhancement and wellbeing can motivate intentional sun exposure, whereas health-preservation motives can encourage protective behaviour.
* '''Protection motivation theory''' helps explain sun-protective behaviour through threat appraisal and coping appraisal, including perceived severity, vulnerability, response efficacy and self-efficacy.
* Effective sun-safety interventions should therefore address both perceptions of health risk and the psychological and social rewards associated with tanning, while strengthening people's confidence in their ability to protect themselves.

== See also ==

* [[w:Health belief model|Health belief model]] (Wikipedia)
* [[w:Protection motivation theory|Protection motivation theory]] (Wikipedia)
* [[w:Sun tanning|Sun tanning]] (Wikipedia)
* [[w:Sunscreen|Sunscreen]] (Wikipedia)
* [[w:Ultraviolet|Ultraviolet radiation]] (Wikipedia)

==References==
{{Hanging indent|1=
Floyd, D. L., Prentice-Dunn, S., &amp; Rogers, R. W. (2000). A meta-analysis of research on protection motivation theory. ''Journal of Applied Social Psychology, 30''(2), 407–429. https://doi.org/10.1111/j.1559-1816.2000.tb02323.x

Hillhouse, J. J., Turrisi, R., &amp; Kastner, M. (2000). Modeling tanning salon behavioral tendencies using appearance motivation, self-monitoring and the theory of planned behavior. ''Health Education Research, 15''(4), 405–414. https://doi.org/10.1093/her/15.4.405

Ingledew, D. K., Ferguson, E., &amp; Markland, D. (2010). Motives and sun-related behaviour. ''Journal of Health Psychology, 15''(1), 8–20. https://doi.org/10.1177/1359105309342292

Jackson, K. M., &amp; Aiken, L. S. (2000). A psychosocial model of sun protection and sunbathing in young women: The impact of health beliefs, attitudes, norms, and self-efficacy for sun protection. ''Health Psychology, 19''(5), 469–478. https://doi.org/10.1037/0278-6133.19.5.469

Jones, J. L., &amp; Leary, M. R. (1994). Effects of appearance-based admonitions against sun exposure on tanning intentions in young adults. ''Health Psychology, 13''(1), 86–90. https://doi.org/10.1037/0278-6133.13.1.86

Leary, M. R., &amp; Jones, J. L. (1993). The social psychology of tanning and sunscreen use: Self-presentational motives as a predictor of health risk. ''Journal of Applied Social Psychology, 23''(17), 1390–1406. https://doi.org/10.1111/j.1559-1816.1993.tb01039.x

Rogers, R. W. (1975). A protection motivation theory of fear appeals and attitude change. ''The Journal of Psychology, 91''(1), 93–114. https://doi.org/10.1080/00223980.1975.9915803
}}

==External links==

* [https://www.cancer.org.au/cancer-information/causes-and-prevention/sun-safety Be SunSmart and sun safety] (Cancer Council Australia)
* [https://www.sunsmart.com.au/ SunSmart] (Cancer Council Victoria)
* [https://www.cancer.org.au/cancer-information/causes-and-prevention/sun-safety/uv-index UV Index] (Cancer Council Australia)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Environment]]
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See also [[OpenStax]]

== Summary ==
''Fundamentals of Nursing'' aligns with the outcomes and competencies of a nursing fundamentals course. It is designed to provide students with the foundational knowledge and practical expertise essential for delivering holistic and patient-centered care. The authors emphasize the application of clinical judgment across diverse healthcare environments, ensuring readiness to deliver high-quality, compassionate care in an inclusive and supportive manner. Composed of 43 chapters, ''Fundamentals of Nursing'' offers an in-depth exploration of the roles and responsibilities of the nursing profession, the healthcare environment, and the application of critical thinking and evidence-based practice.

*[https://openstax.org/details/books/fundamentals-nursing OpenStax Fundamentals of Nursing] (original content). Available as pdf or web view.
*[https://audileo.com/audiobooks/openstax/fundamentals-of-nursing/ OpenStax Fundamentals of Nursing audiobook] Available as audio textbook.

[[Category:OpenStax]]
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      <timestamp>2026-09-02T14:45:30Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>+[[Category:OpenStax]]; +[[Category:Nursing]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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See also [[OpenStax]]

== Summary ==
''Maternal-Newborn Nursing'' introduces students to the concepts and skills related to pregnancy, birth, postpartum, newborn care, reproductive health, and social determinants related to those topics. Written and thoroughly reviewed by experienced nurse educators, the material focuses on patient safety, mental health, and inclusive care, and offers robust real-world scenarios and situational patient education experiences to apply concepts to practice.

* [https://openstax.org/details/books/maternal-newborn-nursing OpenStax Maternal Newborn Nursing] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/maternal-newborn-nursing/ OpenStax Maternal Newborn Nursing audiobook] Available as audio textbook.

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&lt;div align=center&gt;

==Overview==
{{RoundBoxTop|theme=12}}

[[File:Workplace Stress.jpg|Workplace_Stress jpg|right|thumb|200px|'''Figure 1'''. Student experiencing high levels of stress due to poor grades ]]

'''Imagine this...'''

You are a university student who has just received a disappointing grade on an assignment you worked hard on. Your first thought is, &quot;I should have started earlier. I didn't prepare well enough.&quot; As you continue thinking about the result, your thoughts become more personal, &quot;maybe I'm just not smart enough to do well at university.&quot; You begin to feel guilty about how you prepared, ashamed of your performance, and anxious about your next assignment. 

Both reactions involve blaming yourself for the same negative event, but are they psychologically the same? Blaming a specific behaviour may have different emotional consequences from blaming an enduring characteristic of yourself. How might these different forms of self-blame influence your emotions and how you respond to future setbacks?    

As illustrated in Figure 1, negative events can prompt self-evaluation and self-blame, potentially shaping the emotions that follow including high levels of stress when in a similar environment. 


This chapter explores self-blame and how it can influence emotional responses following negative events. Self-blame is a common way of making sense of difficult experiences, particularly when individuals perceive themselves as responsible for what has happened. However, self-blame does not always affect emotions in the same way. The way responsibility is attributed to oneself can shape emotional experiences such as guilt, shame, sadness, and anxiety, and may influence how an individual responds to future challenges. By examining different forms of self-blame and the psychological theories that explain them, tis chapter considers why self-blame may contribute to both adaptive and maladaptive emotional responses. 


{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is self-blame and how does it develop following negative events?
* Why does self-blame influence emotional responses? 
* How are different forms of self-blame associated with emotions such as guilt, shame, sadness and anxiety?
* When might self-blame be adaptive or maladaptive?

{{RoundBoxBottom}}

What Is Self-blame? 
Each chapter should use this standard heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

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&lt;div align=center&gt;

==Overview==
{{RoundBoxTop|theme=12}}

[[File:Workplace Stress.jpg|Workplace_Stress jpg|right|thumb|200px|'''Figure 1'''. Student experiencing high levels of stress due to poor grades ]]

'''Imagine this...'''

You are a university student who has just received a disappointing grade on an assignment you worked hard on. Your first thought is, &quot;I should have started earlier. I didn't prepare well enough.&quot; As you continue thinking about the result, your thoughts become more personal, &quot;maybe I'm just not smart enough to do well at university.&quot; You begin to feel guilty about how you prepared, ashamed of your performance, and anxious about your next assignment. 

Both reactions involve blaming yourself for the same negative event, but are they psychologically the same? Blaming a specific behaviour may have different emotional consequences from blaming an enduring characteristic of yourself. How might these different forms of self-blame influence your emotions and how you respond to future setbacks?    

As illustrated in Figure 1, negative events can prompt self-evaluation and self-blame, potentially shaping the emotions that follow including high levels of stress when in a similar environment. 


This chapter explores self-blame and how it can influence emotional responses following negative events. Self-blame is a common way of making sense of difficult experiences, particularly when individuals perceive themselves as responsible for what has happened. However, self-blame does not always affect emotions in the same way. The way responsibility is attributed to oneself can shape emotional experiences such as guilt, shame, sadness, and anxiety, and may influence how an individual responds to future challenges. By examining different forms of self-blame and the psychological theories that explain them, tis chapter considers why self-blame may contribute to both adaptive and maladaptive emotional responses. 


{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is self-blame and how does it develop following negative events?
* Why does self-blame influence emotional responses? 
* How are different forms of self-blame associated with emotions such as guilt, shame, sadness and anxiety?
* When might self-blame be adaptive or maladaptive?

{{RoundBoxBottom}}


{{RoundBoxTop|theme=12}}
 == Section headings ==
Each chapter should use this standard heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

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&lt;div align=center&gt;

==Overview==
{{RoundBoxTop|theme=12}}

[[File:Workplace Stress.jpg|Workplace_Stress jpg|right|thumb|200px|'''Figure 1'''. Student experiencing high levels of stress due to poor grades ]]

'''Imagine this...'''

You are a university student who has just received a disappointing grade on an assignment you worked hard on. Your first thought is, &quot;I should have started earlier. I didn't prepare well enough.&quot; As you continue thinking about the result, your thoughts become more personal, &quot;maybe I'm just not smart enough to do well at university.&quot; You begin to feel guilty about how you prepared, ashamed of your performance, and anxious about your next assignment. 

Both reactions involve blaming yourself for the same negative event, but are they psychologically the same? Blaming a specific behaviour may have different emotional consequences from blaming an enduring characteristic of yourself. How might these different forms of self-blame influence your emotions and how you respond to future setbacks?    

As illustrated in Figure 1, negative events can prompt self-evaluation and self-blame, potentially shaping the emotions that follow including high levels of stress when in a similar environment. 


This chapter explores self-blame and how it can influence emotional responses following negative events. Self-blame is a common way of making sense of difficult experiences, particularly when individuals perceive themselves as responsible for what has happened. However, self-blame does not always affect emotions in the same way. The way responsibility is attributed to oneself can shape emotional experiences such as guilt, shame, sadness, and anxiety, and may influence how an individual responds to future challenges. By examining different forms of self-blame and the psychological theories that explain them, tis chapter considers why self-blame may contribute to both adaptive and maladaptive emotional responses. 


{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is self-blame and how does it develop following negative events?
* Why does self-blame influence emotional responses? 
* How are different forms of self-blame associated with emotions such as guilt, shame, sadness and anxiety?
* When might self-blame be adaptive or maladaptive?

{{RoundBoxBottom}}


{{RoundBoxTop|theme=12}}
 == What is Self-Blame? ==
=== Focus Question 1: What is self-blame and how does it develop following negative events? ===
Each chapter should use this standard heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]</text>
      <sha1>mn6sgz09rc7ov1uyish4l8r17010le9</sha1>
    </revision>
    <revision>
      <id>2830669</id>
      <parentid>2830667</parentid>
      <timestamp>2026-09-03T04:39:30Z</timestamp>
      <contributor>
        <username>U3281277</username>
        <id>3109857</id>
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&lt;div align=center&gt;

==Overview==
{{RoundBoxTop|theme=12}}

[[File:Workplace Stress.jpg|Workplace_Stress jpg|right|thumb|200px|'''Figure 1'''. Student experiencing high levels of stress due to poor grades ]]

'''Imagine this...'''

You are a university student who has just received a disappointing grade on an assignment you worked hard on. Your first thought is, &quot;I should have started earlier. I didn't prepare well enough.&quot; As you continue thinking about the result, your thoughts become more personal, &quot;maybe I'm just not smart enough to do well at university.&quot; You begin to feel guilty about how you prepared, ashamed of your performance, and anxious about your next assignment. 

Both reactions involve blaming yourself for the same negative event, but are they psychologically the same? Blaming a specific behaviour may have different emotional consequences from blaming an enduring characteristic of yourself. How might these different forms of self-blame influence your emotions and how you respond to future setbacks?    

As illustrated in Figure 1, negative events can prompt self-evaluation and self-blame, potentially shaping the emotions that follow including high levels of stress when in a similar environment. 


This chapter explores self-blame and how it can influence emotional responses following negative events. Self-blame is a common way of making sense of difficult experiences, particularly when individuals perceive themselves as responsible for what has happened. However, self-blame does not always affect emotions in the same way. The way responsibility is attributed to oneself can shape emotional experiences such as guilt, shame, sadness, and anxiety, and may influence how an individual responds to future challenges. By examining different forms of self-blame and the psychological theories that explain them, tis chapter considers why self-blame may contribute to both adaptive and maladaptive emotional responses. 


{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is self-blame and how does it develop following negative events?
* Why does self-blame influence emotional responses? 
* How are different forms of self-blame associated with emotions such as guilt, shame, sadness and anxiety?
* When might self-blame be adaptive or maladaptive?

{{RoundBoxBottom}}


{{RoundBoxTop|theme=12}}
 == What is Self-Blame?==
=== Focus Question 1: What is self-blame and how does it develop following negative events? ===
Each chapter should use this standard heading structure:
* [[#Overview|Overview]]
* 3 to 6 major headings tailored to the topic; can have sub-headings:
** avoid sections with only one sub-heading (use 0 or 2+ sub-headings)
** provide an introductory paragraph before breaking into sub-sections
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]</text>
      <sha1>dwuj8y0uwx1wbid13nxr5jjcf7cwhre</sha1>
    </revision>
    <revision>
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      <parentid>2830669</parentid>
      <timestamp>2026-09-03T05:02:00Z</timestamp>
      <contributor>
        <username>U3281277</username>
        <id>3109857</id>
      </contributor>
      <comment>/* Overview */</comment>
      <origin>2830672</origin>
      <model>wikitext</model>
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&lt;div align=center&gt;

==Overview==
{{RoundBoxTop|theme=12}}

[[File:Workplace Stress.jpg|Workplace_Stress jpg|right|thumb|200px|'''Figure 1'''. Student experiencing high levels of stress due to poor grades ]]

'''Imagine this...'''

You are a university student who has just received a disappointing grade on an assignment you worked hard on. Your first thought is, &quot;I should have started earlier. I didn't prepare well enough.&quot; As you continue thinking about the result, your thoughts become more personal, &quot;maybe I'm just not smart enough to do well at university.&quot; You begin to feel guilty about how you prepared, ashamed of your performance, and anxious about your next assignment. 

Both reactions involve blaming yourself for the same negative event, but are they psychologically the same? Blaming a specific behaviour may have different emotional consequences from blaming an enduring characteristic of yourself. How might these different forms of self-blame influence your emotions and how you respond to future setbacks?    

As illustrated in Figure 1, negative events can prompt self-evaluation and self-blame, potentially shaping the emotions that follow including high levels of stress when in a similar environment. 


This chapter explores self-blame and how it can influence emotional responses following negative events. Self-blame is a common way of making sense of difficult experiences, particularly when individuals perceive themselves as responsible for what has happened. However, self-blame does not always affect emotions in the same way. The way responsibility is attributed to oneself can shape emotional experiences such as guilt, shame, sadness, and anxiety, and may influence how an individual responds to future challenges. By examining different forms of self-blame and the psychological theories that explain them, tis chapter considers why self-blame may contribute to both adaptive and maladaptive emotional responses. 


{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is self-blame and how does it develop following negative events?
* Why does self-blame influence emotional responses? 
* How are different forms of self-blame associated with emotions such as guilt, shame, sadness and anxiety?
* When might self-blame be adaptive or maladaptive?

{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
==   What is Self-Blame? ==
==== Focus Question 1: What is self-blame and how does it develop following negative events? ====

* Self-blame occurs when an individual attributes responsibility or causation for a negative event or outcome to themselves.
* Janoff-Bulman (1979) distinguishes between behavioural self-blame and characterological self-blame, demonstrating that self-blame can involve different explanations for why a negative event occurred.
* Behavioural self-blame focuses on specific and potentially modifiable actions, such as “I didn't study enough”, whereas characterological self-blame focuses on relatively stable characteristics of the self, such as “I'm not intelligent enough” (Janoff-Bulman, 1979).
* Behavioural self-blame may provide a greater sense of control because the perceived cause can potentially be changed, whereas characterological self-blame may reduce perceived control because the cause is viewed as relatively stable (Janoff-Bulman, 1979).
* Research suggests these different attributional styles have different relationships with psychological adjustment, highlighting the importance of distinguishing between forms of self-blame (Anderson et al., 1994; Peterson et al., 1981).
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]</text>
      <sha1>bshfyf0uk2i40lofxxzkjb387vll2uu</sha1>
    </revision>
    <revision>
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      <timestamp>2026-09-03T07:27:59Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>added [[Category:Motivation and emotion/Book/Self-blame]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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&lt;div align=center&gt;

==Overview==
{{RoundBoxTop|theme=12}}

[[File:Workplace Stress.jpg|Workplace_Stress jpg|right|thumb|200px|'''Figure 1'''. Student experiencing high levels of stress due to poor grades ]]

'''Imagine this...'''

You are a university student who has just received a disappointing grade on an assignment you worked hard on. Your first thought is, &quot;I should have started earlier. I didn't prepare well enough.&quot; As you continue thinking about the result, your thoughts become more personal, &quot;maybe I'm just not smart enough to do well at university.&quot; You begin to feel guilty about how you prepared, ashamed of your performance, and anxious about your next assignment. 

Both reactions involve blaming yourself for the same negative event, but are they psychologically the same? Blaming a specific behaviour may have different emotional consequences from blaming an enduring characteristic of yourself. How might these different forms of self-blame influence your emotions and how you respond to future setbacks?    

As illustrated in Figure 1, negative events can prompt self-evaluation and self-blame, potentially shaping the emotions that follow including high levels of stress when in a similar environment. 


This chapter explores self-blame and how it can influence emotional responses following negative events. Self-blame is a common way of making sense of difficult experiences, particularly when individuals perceive themselves as responsible for what has happened. However, self-blame does not always affect emotions in the same way. The way responsibility is attributed to oneself can shape emotional experiences such as guilt, shame, sadness, and anxiety, and may influence how an individual responds to future challenges. By examining different forms of self-blame and the psychological theories that explain them, tis chapter considers why self-blame may contribute to both adaptive and maladaptive emotional responses. 


{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
'''Focus questions'''

Break the sub-title down into three to five [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]]. Align the top-level headings with these focus questions.
* What is self-blame and how does it develop following negative events?
* Why does self-blame influence emotional responses? 
* How are different forms of self-blame associated with emotions such as guilt, shame, sadness and anxiety?
* When might self-blame be adaptive or maladaptive?

{{RoundBoxBottom}}

{{RoundBoxTop|theme=12}}
==   What is Self-Blame? ==
==== Focus Question 1: What is self-blame and how does it develop following negative events? ====

* Self-blame occurs when an individual attributes responsibility or causation for a negative event or outcome to themselves.
* Janoff-Bulman (1979) distinguishes between behavioural self-blame and characterological self-blame, demonstrating that self-blame can involve different explanations for why a negative event occurred.
* Behavioural self-blame focuses on specific and potentially modifiable actions, such as “I didn't study enough”, whereas characterological self-blame focuses on relatively stable characteristics of the self, such as “I'm not intelligent enough” (Janoff-Bulman, 1979).
* Behavioural self-blame may provide a greater sense of control because the perceived cause can potentially be changed, whereas characterological self-blame may reduce perceived control because the cause is viewed as relatively stable (Janoff-Bulman, 1979).
* Research suggests these different attributional styles have different relationships with psychological adjustment, highlighting the importance of distinguishing between forms of self-blame (Anderson et al., 1994; Peterson et al., 1981).
* [[#Conclusion|Conclusion]]
* [[#See also|See also]]
* [[#References|References]]
* [[#External links|External links]]

==Key points==
* For the topic development, provide at least three bullet-points for each heading and sub-heading, including the Overview and Conclusion 
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure; this can be the figure in the scenario
* For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples
* Images must be embedded from [[commons:|Wikimedia Commons]] which hosts free-to-use media such as photos, diagrams, graphs, video, and audio
* Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Embed figures throughout the chapter, starting with the scenario in the Overview section
* Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text)
* Cite each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options:

{{anchor|Scenarios}}
;Scenarios
* Scenarios, case studies, or examples that illustrate concepts in action
* Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* Can be real or fictional; if real, provide citation(s)

{{anchor|Feature box}}
;Feature boxes
* Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
** Quiz questions
** Take-home messages

;Embedded links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia articles (e.g., &quot;An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]&quot;) or
** Related book chapters (e.g., &quot;If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]&quot;)

{{anchor|Tables}}
;Tables
* Use tables to organise and summarise information
* Cite each table at least once in the main text (e.g., see Table 1)
* Tables should be captioned
* [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted

'''Table 1''' 

A Descriptive Caption Explains The Table Contents and its Relevance to the Text e.g., 

The 2 x 2 Johari Window Model Showing the Relationship Between Known/Unknown and Self/Other

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* One or two quiz questions for each main section is better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages
* Ask easy rather than hard questions
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{The purpose of quizzes is to provide an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long and complex quiz questions are recommended:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* Arguably the most important section
* Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking
* For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research
* Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem
* Recommended length: 150 to 330 words

{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
List [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
* [[Motivation and emotion/Assessment/Topic/Checklist|Topic development - Checklist]] (Wikiversity)

{{tip|Suggestions for this section:
* Link to the most relevant internal resources about the topic
* Include the source in parentheses
}}

==References==
Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent.

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper &amp; Row.

}}

{{tip|Suggestions for this section:
* Important aspects of APA referencing style
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case (except first letter and proper names), ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** doi as a URL which is a working hyperlink (i.e., clickable)
** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]:
*** Use &quot;Edit source&quot;
*** &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** dois which aren't clickable as working hyperlinks
** Citing sources that haven't been consulted
}}

==External links==
[[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)

{{tip|Suggestions for this section:
* Link to the most relevant external resources about the topic
* Include the source in parentheses after the link
}}

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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  <page>
    <title>OpenStax College Success</title>
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        <username>Atcovi</username>
        <id>276019</id>
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      <text bytes="951" sha1="40uodxxapgmidbpw4lmk6n0o5xtx681" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;College Success&lt;/big&gt; ==

== Summary ==
OpenStax ''College Success'' is a comprehensive and contemporary resource that serves First Year Experience, Student Success, and College Transition courses. Developed with the support of hundreds of faculty and coordinators, the book addresses the evolving challenges and opportunities of today’s diverse students. Engagement, self-analysis, personal responsibility, and student support are reflected throughout the material. ''College Success'' also includes an array of student surveys and opinion polls, and OpenStax will regularly provide the results to adopting faculty.  

* [https://openstax.org/details/books/college-success OpenStax College Success] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/college-success/ OpenStax College Success audiobook] Available as audio textbook.

[[Category:OpenStax]]</text>
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        <id>276019</id>
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      <comment>added [[Category:Life skills]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="976" sha1="khgyd0udyclgvqdwpj5akm6quyvhclw" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;College Success&lt;/big&gt; ==

== Summary ==
OpenStax ''College Success'' is a comprehensive and contemporary resource that serves First Year Experience, Student Success, and College Transition courses. Developed with the support of hundreds of faculty and coordinators, the book addresses the evolving challenges and opportunities of today’s diverse students. Engagement, self-analysis, personal responsibility, and student support are reflected throughout the material. ''College Success'' also includes an array of student surveys and opinion polls, and OpenStax will regularly provide the results to adopting faculty.  

* [https://openstax.org/details/books/college-success OpenStax College Success] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/college-success/ OpenStax College Success audiobook] Available as audio textbook.

[[Category:OpenStax]]
[[Category:Life skills]]</text>
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    <title>OpenStax Entrepreneurship</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <format>text/x-wiki</format>
      <text bytes="931" sha1="ih33yx4a859g4z28azwuhip65cmdlmg" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Entrepreneurship&lt;/big&gt; ==

== Summary ==
This textbook is intended for use in introductory Entrepreneurship classes at the undergraduate level. Due to the wide range of audiences and course approaches, the book is designed to be as flexible as possible. Theoretical and practical aspects are presented in a balanced manner, and specific components such as the business plan are provided in multiple formats. ''Entrepreneurship'' aims to drive students toward active participation in entrepreneurial roles, and exposes them to a wide range of companies and scenarios.

* [https://openstax.org/details/books/entrepreneurship OpenStax Entrepreneurship] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/entrepreneurship/ OpenStax Entrepreneurship audiobook]  Available as audio textbook.

[[Category:Entrepreneurship]]
[[Category:OpenStax]]</text>
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    <title>OpenStax Introduction to Behavioral Neuroscience</title>
    <ns>0</ns>
    <id>331756</id>
    <revision>
      <id>2830514</id>
      <parentid>2830202</parentid>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
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      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="891" sha1="bhd445jbkzmxhdrakhzszls7owli3tj" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Introduction to Behavioral Neuroscience&lt;/big&gt; ==

== Summary ==
''Introduction to Behavioral Neuroscience'' aligns to the topics and objectives of introductory behavioral neuroscience courses taught in psychology, biology, neuroscience, and similar departments. This offering is intended for undergraduates with no presumed college-level science coursework, and presents the foundational principles of brain-behavior-environment interactions.

* [https://openstax.org/details/books/introduction-behavioral-neuroscience OpenStax Introduction to Behavioral Neuroscience] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/introduction-to-behavioral-neuroscience/ OpenStax Introduction to Behavioral Neuroscience audiobook] Available as audio textbook.

[[Category:Neuroscience]]
[[Category:OpenStax]]</text>
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    <title>OpenStax Microbiology</title>
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    <revision>
      <id>2830524</id>
      <parentid>2830201</parentid>
      <timestamp>2026-09-02T14:45:43Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>added [[Category:OpenStax]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <format>text/x-wiki</format>
      <text bytes="941" sha1="lhvi5669k49a63vaeg8gwfm0p2ipik1" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Microbiology&lt;/big&gt; ==

== Summary ==
''Microbiology'' covers the scope and sequence requirements for a single-semester microbiology course for non-majors. The book presents the core concepts of microbiology with a focus on applications for careers in allied health. The pedagogical features of the text make the material interesting and accessible while maintaining the career-application focus and scientific rigor inherent in the subject matter. ''Microbiology''’s art program enhances students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs.

* [https://openstax.org/details/books/microbiology OpenStax Microbiology] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/microbiology/ OpenStax Microbiology audiobook] Available as audio textbook.

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    <title>OpenStax Medical Surgical Nursing</title>
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        <id>2888076</id>
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See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Medical Surgical Nursing&lt;/big&gt; ==

== Summary ==
''Medical-Surgical Nursing'' is a comprehensive and engaging resource for nursing students, equipping them with the essential knowledge and skills required to provide high-quality care. Building on the foundational knowledge and skills acquired in previous nursing courses, this textbook expands students’ learning with concepts in medical-surgical nursing. The offering provides the background to ensure that nursing students are well-versed in both the science and art of nursing, capable of critical thinking, clinical judgment, and compassionate care.

* [https://openstax.org/details/books/medical-surgical-nursing OpenStax Medical-Surgical Nursing] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/medical-surgical-nursing/ OpenStax Medical-Surgical Nursing audiobook] Available as audio textbook.

[[Category:Nursing]]</text>
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    <title>OpenStax Pharmacology for Nurses</title>
    <ns>0</ns>
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    <revision>
      <id>2830477</id>
      <parentid>2830450</parentid>
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      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
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      <text bytes="1001" sha1="nkzl6sf2xe0g9sk0z03yjz314de6uxh" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Pharmacology for Nurses&lt;/big&gt; ==

== Summary ==
''Pharmacology for Nurses'' is intended for nursing students in an introductory program. It provides a fundamental understanding of the therapeutic use of drugs, so the nurse can provide safe and effective care to the client. Along with a discussion of each body system, the text also reviews the pathophysiology of various disease processes and medications used in treatment. The table of contents presents 40 chapter topics, organized into 11 units. The first unit, consisting of 3 chapters, provides a broad overview of pharmacology, with the following 10 units focused on specific body systems.

* [https://openstax.org/details/books/pharmacology OpenStax Pharmacology for Nurses] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/pharmacology-for-nurses/ OpenStax Pharmacology for Nurses audiobook] Available as audio textbook.

[[Category:Pharmacology]]</text>
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      <text bytes="1023" sha1="1pi6bbpyr2gr7mv6rudd2dwfcer82sg" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; &lt;big&gt;Pharmacology for Nurses&lt;/big&gt; ==

== Summary ==
''Pharmacology for Nurses'' is intended for nursing students in an introductory program. It provides a fundamental understanding of the therapeutic use of drugs, so the nurse can provide safe and effective care to the client. Along with a discussion of each body system, the text also reviews the pathophysiology of various disease processes and medications used in treatment. The table of contents presents 40 chapter topics, organized into 11 units. The first unit, consisting of 3 chapters, provides a broad overview of pharmacology, with the following 10 units focused on specific body systems.

* [https://openstax.org/details/books/pharmacology OpenStax Pharmacology for Nurses] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/pharmacology-for-nurses/ OpenStax Pharmacology for Nurses audiobook] Available as audio textbook.

[[Category:Pharmacology]]
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  <page>
    <title>Universal Bibliography/Culture</title>
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This part of the [[Universal Bibliography]] is a bibliography of culture.

*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

==Japanese==
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Joseph Roggendorf (ed). Studies in Japanese Culture. Sophia University. Tokyo. 1963. [https://books.google.co.uk/books?id=VnBwAAAAMAAJ]
**Joseph Roggendorf (ed). Studies in Japanese Culture: Tradition and Experiment. Sophia University. Tokyo. 2nd Ed. 2nd printing. 1965. [https://books.google.co.uk/books?id=GHJwAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

==Korean==
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

==Latin American==
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

[[Category:Culture]]</text>
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      <text bytes="13129" sha1="apj64tdodsrvr65bs6iyp8jf7cbehg4" xml:space="preserve">{{Bibliography}}

This part of the [[Universal Bibliography]] is a bibliography of culture.

*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

==Japanese==
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Joseph Roggendorf (ed). Studies in Japanese Culture. Sophia University. Tokyo. 1963. [https://books.google.co.uk/books?id=VnBwAAAAMAAJ]
**Joseph Roggendorf (ed). Studies in Japanese Culture: Tradition and Experiment. Sophia University. Tokyo. 2nd Ed. 2nd printing. 1965. [https://books.google.co.uk/books?id=GHJwAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Modern
*Yoshio Sugimoto. The Cambridge Companion to Modern Japanese Culture. (Cambridge Companions to Culture). 2009. ISBN 9780521880473. [https://books.google.co.uk/books?id=s80AAwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Leith Morton. Modern Japanese Culture: The Insider View. Oxford University Press. 2003. ISBN 0195540891. [https://books.google.com/books?id=1XZgQgAACAAJ]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

==Korean==
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

==Latin American==
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

[[Category:Culture]]</text>
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This part of the [[Universal Bibliography]] is a bibliography of culture.

*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

==Japanese==
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Joseph Roggendorf (ed). Studies in Japanese Culture. Sophia University. Tokyo. 1963. [https://books.google.co.uk/books?id=VnBwAAAAMAAJ]
**Joseph Roggendorf (ed). Studies in Japanese Culture: Tradition and Experiment. Sophia University. Tokyo. 2nd Ed. 2nd printing. 1965. [https://books.google.co.uk/books?id=GHJwAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Daniel Sosnoski. Introduction to Japanese Culture. Tuttle. 1996. [https://books.google.co.uk/books?id=kJ3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*T Smibert and C Burns. Art, Nature and Life: An Introduction to Japanese Culture. Educational Media Australia. Melbourne. 1979. Bibliography: [https://books.google.co.uk/books?id=tbYlAQAAIAAJ].

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Modern
*Yoshio Sugimoto. The Cambridge Companion to Modern Japanese Culture. (Cambridge Companions to Culture). 2009. ISBN 9780521880473. [https://books.google.co.uk/books?id=s80AAwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Leith Morton. Modern Japanese Culture: The Insider View. Oxford University Press. 2003. ISBN 0195540891. [https://books.google.com/books?id=1XZgQgAACAAJ]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

==Korean==
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

==Latin American==
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

[[Category:Culture]]</text>
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      <text bytes="13661" sha1="co6x84ntgc65ab5uv9zf0w7bgmt7ai7" xml:space="preserve">{{Bibliography}}

This part of the [[Universal Bibliography]] is a bibliography of culture.

*Terry Eagleton. Culture. Yale University Press. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ben Highmore. Culture. Routledge. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks. Culture. Routledge. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=VcGHAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Chris Jenks (ed). Culture: Critical Concepts in Sociology. 2003. [https://books.google.co.uk/books?id=VIEbAQAAMAAJ vol 1]. [https://books.google.co.uk/books?id=iIvSvY7HWv8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2]. [https://books.google.co.uk/books?id=ZF1Hk2HytAoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 3].
*Craig Calhoun (ed). Culture. (Comparative Social Research: A Research Annual, Volume 11: 1989). JAI Press. 1989. [https://books.google.co.uk/books?id=qvpWAAAAYAAJ]

*Crane. The Production of Culture: Media and the Urban Arts. (Foundations of Popular Culture, vol 1). SAGE Publications. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Richard A Peterson. The Production of Culture. (SAGE Contemporary Social Science Issues 33). SAGE Publications. 1976. [https://books.google.co.uk/books?id=DILWAAAAMAAJ].
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. Columbia University Press. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Zygmunt Bauman. Culture as Praxis. 1973. Sage. 1999. [https://books.google.co.uk/books?id=7yqa0-s6N24C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Material culture
*Thomas J Schlereth (ed). Material Culture: A Research Guide. University Press of Kansas. 1985. [https://books.google.co.uk/books?id=Ig9PAAAAMAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert DuPlessis. Material Culture. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=Y-p4acTurWQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

Asian

Periodicals, Asian
*Asian Culture Quarterly. Asian Cultural Center, Taipei [https://books.google.co.uk/books?id=4PYtAQAAIAAJ]
*Asian Culture: A Quarterly Review. Vietnamese Association for Asian Cultural Relations (Hội Việt Nam Nghiên Cứu Liên Lạc Văn Hoá Á Châu). [https://books.google.co.uk/books?id=g9YbAAAAMAAJ]
*Journal of Asian Culture. Graduate Students in Asian Studies at UCLA. [https://books.google.co.uk/books?id=-HBtAAAAMAAJ]

Asian popular culture
*Anthony YH Fung (ed). Asian Popular Culture: The Global (Dis)continuity. 2013. [https://books.google.co.uk/books?id=Htqw-BHJlnwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lorna Fitzsimmons and John A Lent (eds). Asian Popular Culture in Transition. (Routledge Contemporary Asia Series). 2013. [https://books.google.co.uk/books?id=UMZ7eH6dDxkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John A Lent and Lorna Fitzsimmons (eds). Asian Popular Culture: New, Hybrid, and Alternate Media. Lexington Books. 2013. [https://books.google.co.uk/books?id=tG6AEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Yeojin Kim, Dharshani Lakmali Jayasinghe, Hiba Aleem and Karen A Ritzenhoff (eds). Contemporary Asian Popular Culture. Palgrave Macmillan. [https://books.google.co.uk/books?id=G0g9EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=fZo3EQAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false vol 2].

East Asian
*Xiaobing Tang and Stephen Snyder (eds). In Pursuit Of Contemporary East Asian Culture. [https://books.google.co.uk/books?id=dXekDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Popular culture
*Koichi Iwabuchi, Eva Tsai and Chris Berry (eds). Routledge Handbook of East Asian Popular Culture. 2017. [https://books.google.co.uk/books?id=lCMlDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] 
*Seok-Kyeong Hong and Dal Yong Jin (eds). Transnational Convergence of East Asian Pop Culture. 2021. [https://books.google.co.uk/books?id=hbsTEAAAQBAJ&amp;pg=PA2000#v=onepage&amp;q&amp;f=false]

Periodicals
*East Asian Cultural Studies. The Centre for East Asian Cultural Studies, Tokyo. [https://books.google.co.uk/books?id=4OTUAAAAMAAJ]

==Japanese==
*Paul Varley. Japanese Culture. 1984. 4th Ed: 2000: [https://books.google.co.uk/books?id=BvUEzBin61AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 
**Japanese Culture: A Short History. 1973. 1977. [https://books.google.co.uk/books?id=bas6AAAAMAAJ]
*Ruth Benedict. The Chrysanthemum and the Sword: Patterns of Japanese Culture. 1946. A Mariner Book. 2005. [https://books.google.co.uk/books?id=R7NpvfYsmU0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Robert J Smith and Richard K Beardsley (eds). Japanese Culture: Its Development and Characteristics. 1963. [https://books.google.co.uk/books?id=gZFkfPrNErEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Eiichiro Ishida. Japanese Culture: A Study of Origins and Characteristics. (Translated by Teruko Kachi). 1974. [https://books.google.com/books?id=gztxAAAAMAAJ]
*Pictorial Encyclopedia of Japanese Culture: The Soul and Heritage of Japan. Gakken. 1987. [https://books.google.com/books?id=cNomAQAAMAAJ]
*Setsuko Kojima and Gene A Crane. A Dictionary of Japanese Culture. Chopmen Publishers. [https://books.google.co.uk/books?id=ms_rAAAAMAAJ]
*Joseph Roggendorf (ed). Studies in Japanese Culture. Sophia University. Tokyo. 1963. [https://books.google.co.uk/books?id=VnBwAAAAMAAJ]
**Joseph Roggendorf (ed). Studies in Japanese Culture: Tradition and Experiment. Sophia University. Tokyo. 2nd Ed. 2nd printing. 1965. [https://books.google.co.uk/books?id=GHJwAAAAMAAJ]
*Studies in Japanese Culture. University of Michigan Press. [https://books.google.co.uk/books?id=kxAOAQAAMAAJ]
*Murakami Hyôe and Edward G Seidensticker (eds). Guides to Japanese Culture. Japan Culture Institute. 1977. ISBN 0-87040-403-2 [https://books.google.co.uk/books?id=xiQhhOvtVJEC]
*Chikio Hayashi and Yasumasa Kuroda. Japanese Culture in Comparative Perspective. Praeger. Westport, Connecticut. 1997.  [https://books.google.com/books?id=tTzDEAAAQBAJ]
*Donald H Shively (ed). Tradition and Modernization in Japanese Culture. Princeton University Press. 1971. Paperback. 1976. [https://books.google.co.uk/books?id=MnN9BgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Daniel Sosnoski. Introduction to Japanese Culture. Tuttle. 1996. [https://books.google.co.uk/books?id=kJ3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*T Smibert and C Burns. Art, Nature and Life: An Introduction to Japanese Culture. Educational Media Australia. Melbourne. 1979. Bibliography: [https://books.google.co.uk/books?id=tbYlAQAAIAAJ].

Cultural history
*Shunsuke Tsurumi. A Cultural History of Postwar Japan 1945-1980. Iwanami Shoten. Tokyo. 1984.  Kegan Paul International. 1987. Routledge. 2009. [https://books.google.co.uk/books?id=6HYsBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*GB Sansom. Japan: A Short Cultural History. 1931. Revised Ed. [https://books.google.co.uk/books?id=ZOpkCwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Dougill. Kyoto: A Cultural History. (Cityscapes). Oxford University Press. 2006. [https://books.google.co.uk/books?id=rhMTDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Nippon No Bunka

[日本の文化 = Nippon No Bunka = Nihon No Bunka = Japanese culture]
*Genshoku Nihon No Bunka (Japanese: 原色日本の文化). Shogakukan (小学館). 1968. [https://books.google.co.uk/books?id=FKwdAQAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN15289454].
*日本文化提要 - Guides to Japanese Culture. 日本文化研究所編集局. 1977. [https://books.google.co.uk/books?id=8noNAQAAMAAJ] [https://books.google.co.uk/books?id=hXcEAAAAMAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/BN02209055].

Contemporary
*Sandra Buckley. The Encyclopedia of Contemporary Japanese Culture. Routledge. 2002. [https://books.google.co.uk/books?id=3ZKFAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Wtkm3O3nWXkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Roger J Davies and Osamu Ikeno (eds). Japanese Mind: Understanding Contemporary Japanese Culture. Tuttle Publishing. 2002. [https://books.google.co.uk/books?id=rADRAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Modern
*Yoshio Sugimoto. The Cambridge Companion to Modern Japanese Culture. (Cambridge Companions to Culture). 2009. ISBN 9780521880473. [https://books.google.co.uk/books?id=s80AAwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Leith Morton. Modern Japanese Culture: The Insider View. Oxford University Press. 2003. ISBN 0195540891. [https://books.google.com/books?id=1XZgQgAACAAJ]

Popular culture
*Hidetoshi Kato (ed). Japanese Popular Culture. Charles E Tuttle Company. Rutland, Vermont. Tokyo. 1959. [https://books.google.com/books?id=HkVxAAAAMAAJ]
**Richard Gid Powers, Hidetoshi Kato and Bruce Stronach (eds). Handbook of Japanese Popular Culture. Greenwood Press. Westport, Connecticut. 1989. [https://books.google.co.uk/books?id=tuRwAAAAMAAJ] 
*Alisa Freedman and Toby Slade (eds). Introducing Japanese Popular Culture. [https://books.google.co.uk/books?id=UBFFDwAAQBAJ&amp;pg=PA1963#v=onepage&amp;q&amp;f=false]. Alisa Freedman (ed). 2nd Ed: 2023: [https://books.google.co.uk/books?id=WuC0EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Timothy J Craig (ed). Japan Pop! Inside the World of Japanese Popular Culture. ME Sharpe. 2000. [https://books.google.co.uk/books?id=Vvw5WQ0crLoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*E Taylor Atkins. A History of Popular Culture in Japan: From the Seventeenth Century to the Present. Bloomsbury Academic. 2017: [https://books.google.co.uk/books?id=DK41DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2nd Ed: 2023: [https://books.google.co.uk/books?id=hGuHEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Douglas Slaymaker. A Century of Popular Culture in Japan. [https://books.google.co.uk/books?id=H_dxAAAAMAAJ]
*John Whittier Treat. Contemporary Japan and Popular Culture. University of Hawaii Press. 1996. [https://books.google.co.uk/books?id=iuYwAQAAIAAJ]
*D P Martinez (ed). The Worlds of Japanese Popular Culture: Gender, Shifting Boundaries and Global Cultures. Cambridge University Press. 1998. [https://books.google.co.uk/books?id=6mqM8m-sJY4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Nissim Kadosh Otmazgin. Regionalizing Culture: The Political Economy of Japanese Popular Culture in Asia. University of Hawaii Press. 2014. [https://books.google.co.uk/books?id=vVgEEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*William W Kelly (ed). Fanning the Flames: Fans and Consumer Culture in Contemporary Japan. 2004. [https://books.google.co.uk/books?id=Y_LmaLo4190C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Arts and culture
*Nelly Delay. Art and Culture of Japan. (Discoveries). Harry N Abrams. 1999.  [https://books.google.co.uk/books?id=VlyAILUBFLYC]
*Stephen Addiss, Gerald Groemer and J Thomas Rimer (eds). Traditional Japanese Arts and Culture: An Illustrated Sourcebook. University of Hawaii Press. 2006. [https://books.google.co.uk/books?id=TZcBEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Leisure
*Sepp Linhart and Sabine Fruhstuck (eds). The Culture of Japan as Seen through Its Leisure. State University of New York Press. 1998. [https://books.google.co.uk/books?id=Ev_n7VaEpUoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Culture and customs
*Noriko Kamachi. Culture and Customs of Japan. (Culture and Customs of Asia: ISSN 1097-0738). 1999. [https://books.google.co.uk/books?id=2cvWAAAAMAAJ]

Series
*Japanese Culture in the Meiji Era

==Korean==
*John H Koo and Andrew C Nahm (eds). An Introduction to Korean Culture. Hollym. [https://books.google.co.uk/books?id=975xAAAAMAAJ]
*Joanne Miyang Cho and Lee M Roberts (eds). Korean Culture in the Global Age: K-Pop, K-Drama, K-Film, and K-Literature. 2025. [https://books.google.co.uk/books?id=2ftgEQAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Richard Saccone. The Business of Korean Culture. Hollym. 1994. [https://books.google.co.uk/books?id=7ji2AAAAIAAJ]

Periodicals
*Korean Culture. Korean Cultural Service. ISSN 0270-1618.

==Latin American==
History
*Pedro Henríquez Ureña. A Concise History of Latin American Culture. Frederick A Praeger. 1966. [https://books.google.co.uk/books?id=yZxoAAAAMAAJ]

Contemporary
*C Gail Guntermann (ed). Contemporary Latin American Culture: Unity and Diversity. 1984. [https://books.google.co.uk/books?id=FjUcAQAAIAAJ]
*George M Foster (ed). Readings in Contemporary Latin American Culture: An Anthropological Sourcebook. Selected Academic Readings. [https://books.google.co.uk/books?id=a89AAAAAIAAJ]

Modern
*John King (ed). The Cambridge Companion to Modern Latin American Culture. 2004. [https://books.google.co.uk/books?id=BE30cHBvjM8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Anthropology
*Emilio Willems. Latin American Culture: An Anthropological Synthesis. Harper &amp; Row. [https://books.google.co.uk/books?id=mn8WAAAAYAAJ]

Teaching
*Gloria Contreras. Latin American Culture Studies: Information and Materials for Teaching about Latin America. Institute of Latin American Studies, University of Texas at Austin. [https://books.google.co.uk/books?id=kn9qAAAAMAAJ]

Cultural studies
*Ana del Sarto, Alicia Ríos and Abril Trigo (eds). The Latin American Cultural Studies Reader. 2004.[https://books.google.co.uk/books?id=NW1CXsOKRc8C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stephen Hart and Richard Young. Contemporary Latin American Cultural Studies. Hodder Arnold. 2003. Routledge. 2014. [https://books.google.co.uk/books?id=o7TpAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

[[Category:Culture]]</text>
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== Rotations ==
The [[wikipedia:Rotations in 4-dimensional Euclidean space#Isoclinic_rotations|isoclinic rotations]] of the convex [[wikipedia:regular 4-polytope|regular 4-polytopes]] are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[wikipedia:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[wikipedia:SO(4)#Double_rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[wikipedia:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[wikipedia:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[wikipedia:Euclidean geometry#Higher_dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[wikipedia:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell.

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[wikipedia:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell.

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[wikipedia:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[wikipedia:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[wikipedia:torsion of a curve|torsion]].... The fibres of this [[wikipedia:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[wikipedia:Clifford parallel|Clifford parallels]].{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm?

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;...</text>
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    <title>David Ricardo/An Essay on Profits</title>
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      <text bytes="3405" sha1="0aj46mgqv0hvnjrkdua0nz923odxoen" xml:space="preserve">This resource gives information on [[David Ricardo]]'s ''An Essay on the Influence of a Low Price of Corn on the Profits of Stock'' (often abridged to &quot;(An) Essay on Profits'').

{|
| {{pad|600px}}
|“It is doubtful whether it is possible&lt;br /&gt;to understand the Principles without&lt;br /&gt;knowledge of the contents of the Essay”&lt;br /&gt;(O’Brien 1975, p. 139).
|}

Two years before the publication of the first edition of his Magnum Opus, ''On the Principles of Political Economy and Taxation'' (often abridged to ''(The) Principles''), in 1815, Ricardo published a pamphlet entitled ''An Essay on the Influence of a Low Price of Corn on the Profits of Stock''.&lt;br /&gt;
Full title:  An Essay on the Influence of a Low Price of Corn on the Profits of Stock; Shewing the Inexpediency of Restrictions on Importation; With Remarks on Mr. Malthus' Two Last Publications: &quot;An Inquiry into the Nature and Progress of Rent;&quot; and &quot;The Grounds of an Opinion on the Policy of Restricting the Importation of Foreign Corn.&quot;&lt;ref name=Essay1&gt;{{cite book 
| last = Ricardo | first = David
| date = 1815
| title =  An Essay on the Influence of a Low Price of Corn on the Profits of Stock
| location = London
| publisher = John Murray
}}&lt;br /&gt;
We will use two important editions:
* 1815: 2nd edition, online available at wikisource: [[:s:en:The Essay on Profits|The Essay on Profits (2nd ed.)]]
* 1962: in the &quot;Works and Correspondence of David Ricardo,'' edited by Piero Sraffa, with the collaboration of Maurice Dobb, Volume 4 (Pamphlets and Papers, 1815-1823), pp. 9-41. With note by P. Sraffa, p. 1-8. available to borrow at [https://archive.org/details/workscorresponde0004rica The Internet Archive] after (free) subscription.&lt;/ref&gt;

Opinions among Ricardo scholars differ as to the precise significance of this “Essay on Profits”. Among other things, there are subtle differences regarding the interpretation of the main variable under investigation,&lt;ref&gt;According to Hollander 1973, p. 273, “the main variable under investigation in the Essay (…) is the general rate of profit”.&lt;br /&gt;According to Stigler 1952, pp. 201–202, the Essay contains “two main elements of the Ricardian system: the theory of rent and the dominant influence of diminishing returns in agriculture upon the rate of profit.”&lt;br /&gt;According to Eatwell 1975, p. 186, the Essay is “the first complete statement of a theory of distribution based on the concept of surplus; that is to say, of a deductive relationship between wages and profits”.&lt;br /&gt;According to Stigler 1952, pp. 201–202, the Essay contains “two main elements of the Ricardian system: the theory of rent and the dominant influence of diminishing returns in agriculture upon the rate of profit.”&lt;/ref&gt; and there are disagreements about the relationship between the “Essay” and the “Principles” &lt;ref&gt;Compare, for example, the discussion between Hollander and Eatwell on the significance of the &quot;corn model&quot; in Hollander 1973 and 1975, and Eatwell 1975.&lt;/ref&gt;

In the ''Essay on Profits'' Ricardo presents a comprehensive theory of ground rent within the context of a &quot;corn model&quot;.&lt;ref&gt;This model is set out on pages 10 to 18 of the ‘Essay on Profits’. From page 18 onwards, the grain is (suddenly) assigned a price.&lt;/ref&gt;

== Footnotes ==
{{references}}

== Sources ==
* Eatwell 1975
* Hollander 1973
* Hollander 1975
* O'Brien 1975
* Stigler 1952</text>
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|“It is doubtful whether it is possible&lt;br /&gt;to understand the Principles without&lt;br /&gt;knowledge of the contents of the Essay.”&lt;br /&gt;(O’Brien 1975, p. 139).
|}

This resource gives information on '''[[David Ricardo]]'s''' '''''An Essay on the Influence of a Low Price of Corn on the Profits of Stock''''' (often abridged to '''''(An) Essay on Profits''''').

Two years before the publication of the first edition of his Magnum Opus, ''On the Principles of Political Economy and Taxation'' (often abridged to ''(The) Principles''), in 1815, Ricardo published a pamphlet entitled ''An Essay on the Influence of a Low Price of Corn on the Profits of Stock''.

Full title:  An Essay on the Influence of a Low Price of Corn on the Profits of Stock; Shewing the Inexpediency of Restrictions on Importation; With Remarks on Mr. Malthus' Two Last Publications: &quot;An Inquiry into the Nature and Progress of Rent;&quot; and &quot;The Grounds of an Opinion on the Policy of Restricting the Importation of Foreign Corn.&quot;&lt;ref name=Essay1&gt;{{cite book 
| last = Ricardo | first = David
| date = 1815
| title =  An Essay on the Influence of a Low Price of Corn on the Profits of Stock
| location = London
| publisher = John Murray
}}&lt;br /&gt;
We will use two important editions:
* 1815: 2nd edition, online available at [[Image:wikisource-logo.svg|16x16px]] Wikisource: [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock|An Essay on Profits (2nd ed.)]]
* 1962: in the ''Works and Correspondence of David Ricardo,'' edited by Piero Sraffa, with the collaboration of Maurice Dobb, Volume 4 (Pamphlets and Papers, 1815-1823), pp. 9-41. With note by P. Sraffa, p. 1-8. available to borrow at [https://archive.org/details/workscorresponde0004rica The Internet Archive] after (free) subscription.&lt;/ref&gt;

Opinions among Ricardo scholars differ as to the precise significance of this “Essay on Profits”. Among other things, there are subtle differences regarding the interpretation of the main variable under investigation,&lt;ref&gt;According to Hollander 1973, p. 273, “the main variable under investigation in the Essay (…) is the general rate of profit”.&lt;br /&gt;According to Stigler 1952, pp. 201–202, the Essay contains “two main elements of the Ricardian system: the theory of rent and the dominant influence of diminishing returns in agriculture upon the rate of profit.”&lt;br /&gt;According to Eatwell 1975, p. 186, the Essay is “the first complete statement of a theory of distribution based on the concept of surplus; that is to say, of a deductive relationship between wages and profits”.&lt;/ref&gt; and there are disagreements about the relationship between the “Essay” and the “Principles”.&lt;ref&gt;Compare, for example, the discussion between Hollander and Eatwell on the significance of the &quot;corn model&quot; in Hollander 1973 and 1975, and Eatwell 1975.&lt;/ref&gt;

In the ''Essay on Profits'' Ricardo presents a comprehensive theory of ground rent within the context of a &quot;corn model&quot;.&lt;ref&gt;This model is set out on pages 10 to 18 of the ''Essay on Profits.'' From page 18 onwards, the grain is (suddenly) assigned a price.&lt;/ref&gt;

== Footnotes ==
{{references}}

== Sources ==
* {{cite journal
| last = Eatwell | first = John | author-link = [[:w:en:John Eatwell]]
| date = 1975
| title = The Interpretation of Ricardo's Essay on Profits
| journal = Economica
| volume = XLII
| pp = 182–187
}}
* Hollander 1973
* Hollander 1975
* O'Brien 1975
* Stigler 1952</text>
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    <title>Introduction to thermodynamics/Chapter one review</title>
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      <comment>Created page with &quot;The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.  Several useful forms of the first law:&quot;</comment>
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      <text bytes="173" sha1="ob3bc6usrbv3vl8c9s1v1vvvzv0syac" xml:space="preserve">The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.

Several useful forms of the first law:</text>
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      <text bytes="157" sha1="ktr4jp9nv506v460lr7mgfcinq8e7dp" xml:space="preserve">Thermodynamics is the study of heat and work. The physical laws of thermodynamics are a result of experimental validation of the nature of reality around us.</text>
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    <title>User:Mitchal Dichter/Pre-University Mathematics/Exponents/</title>
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      <text bytes="1525" sha1="27u2c33w18z8rwynohvfto5izm13frr" xml:space="preserve">Exponents are a mathematical operation. In the simplest case, exponents are a shorter way to write repeated multiplication. In the example of &lt;math&gt;7&lt;/math&gt; multiplied by itself &lt;math&gt;5&lt;/math&gt; times, &lt;math&gt;7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 7^{5}&lt;/math&gt; where &lt;math&gt;7&lt;/math&gt; is called the '''base''' of the exponent and &lt;math&gt;5&lt;/math&gt; is called the '''power''' of the exponent. The common ways to write in words and say &lt;math&gt;7^{5}&lt;/math&gt; are &quot;seven to the power 5&quot; and &quot;seven to the fifth power&quot;.

==Relationship to Units of Measurement==

A common situation people first see exponents is for length, area, and volume. Using meters as the unit of measurement,
* meters &lt;math&gt;m = m^{1}&lt;/math&gt;
* meters squared &lt;math&gt;m \cdot m = m^{2}&lt;/math&gt;
* meters cubed &lt;math&gt;m \cdot m \cdot m = m^{3}&lt;/math&gt;.
Writing &lt;math&gt;m^{1}&lt;/math&gt; is uncommon but is useful for describing exponents in the context of maths. The &lt;math&gt;2&lt;/math&gt; and &lt;math&gt;3&lt;/math&gt; go with a square in two dimensions, and a cube in three dimensions.

All the common ways to write in words and say &lt;math&gt;m^{1}&lt;/math&gt;, &lt;math&gt;m^{2}&lt;/math&gt;, and &lt;math&gt;m^{3}&lt;/math&gt; are:
* &quot;meters to the first power&quot; or &quot;meters to the power 1&quot;
* &quot;meters to the second power&quot; or &quot;meters to the power 2&quot; or &quot;meters squared&quot;
* &quot;meters to the third power&quot; or &quot;meters to the power 3&quot; or &quot;meters cubed&quot;

Another situation people first see exponents is speed. Speed can be written as a fraction &lt;math&gt;\frac{m}{s}&lt;/math&gt;, and sometimes &lt;math&gt;m s^{-1}&lt;/math&gt; using a negative exponent on the seconds.</text>
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      <text bytes="6686" sha1="qw2pgzke1qlla7h1dnnx2t3ms7lci2t" xml:space="preserve">The exponent operation, also known as exponentiation, is a mathematical operation. In the simplest case, exponents are a shorter way to write repeated multiplication. In the example of &lt;math&gt;7&lt;/math&gt; multiplied by itself &lt;math&gt;5&lt;/math&gt; times, &lt;math&gt;7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 7^{5}&lt;/math&gt; where &lt;math&gt;7&lt;/math&gt; is called the '''base''' of the exponent and &lt;math&gt;5&lt;/math&gt; is called the '''power'''. The common ways to write in words and say &lt;math&gt;7^{5}&lt;/math&gt; are &quot;seven to the power 5&quot; and &quot;seven to the fifth power&quot;.

Unfortunately, the word word '''exponent''' is also a name for the '''power''', as in &lt;math&gt;{5}&lt;/math&gt; is the '''exponent''' in &lt;math&gt;7^{5}&lt;/math&gt;.

==Relationship to Units of Measurement==

A common situation people first see exponents is for length, area, and volume. Using meters as the unit of measurement,
* meters &lt;math&gt;m = m^{1}&lt;/math&gt;
* meters squared &lt;math&gt;m \cdot m = m^{2}&lt;/math&gt;
* meters cubed &lt;math&gt;m \cdot m \cdot m = m^{3}&lt;/math&gt;
Writing &lt;math&gt;m^{1}&lt;/math&gt; is uncommon but is useful for describing exponents in the context of maths. The &lt;math&gt;2&lt;/math&gt; and &lt;math&gt;3&lt;/math&gt; go with a square in two dimensions and a cube in three dimensions.

All the common ways to write in words and say &lt;math&gt;m^{1}&lt;/math&gt;, &lt;math&gt;m^{2}&lt;/math&gt;, and &lt;math&gt;m^{3}&lt;/math&gt; are:
* &quot;meters to the first power&quot; or &quot;meters to the power 1&quot;
* &quot;meters to the second power&quot; or &quot;meters to the power 2&quot; or &quot;meters squared&quot;
* &quot;meters to the third power&quot; or &quot;meters to the power 3&quot; or &quot;meters cubed&quot;

Another situation people first see exponents is speed. Speed can be written as a fraction &lt;math&gt;\frac{m}{s}&lt;/math&gt;, and sometimes &lt;math&gt;m s^{-1}&lt;/math&gt; using a negative exponent on the seconds.

==Positive Exponents==
We saw how positive exponents are used for length, area, and volume above with the unit of length meters &lt;math&gt;m&lt;/math&gt;, and the same rules apply when meters &lt;math&gt;m&lt;/math&gt; is replaced with a number like 7.
* &lt;math&gt;7 = 7^{1}&lt;/math&gt;
* 7 squared &lt;math&gt;= 7 \cdot 7 = 7^{2}&lt;/math&gt;
* &quot;7 to the third power&quot; and &quot;7 to the power 3&quot; and &quot;7 cubed&quot; all equal &lt;math&gt;7^{3}&lt;/math&gt;

==How to Type an Exponent on a Computer Keyboard==
caret symbol ^

==Practice with Positive Exponents==
Write all these expressions as a base number to the power of an exponent. &lt;math&gt;b^{p}&lt;/math&gt;
# &lt;math&gt;4 \cdot 4 \cdot 4&lt;/math&gt;
# &lt;math&gt;5&lt;/math&gt;
# &lt;math&gt;7 \cdot 7 \cdot 7 \cdot 7 \cdot 7&lt;/math&gt;
# 13 to the fourth power
# 99 to the power 78
# 9 squared
# 57 to the power 8
# 33 cubed
# 2 to the 17th power
The solutions are at the bottom of the page.

==Real-World Example: Uh-Oh! You're part of a [https://en.wikipedia.org/wiki/Pyramid_scheme Pyramid Scheme]!==
This example is to show how large a number can become from from increasing the exponent just a little.

==Real-World Example: Computer Memory Names==


==Parentheses and Positive Exponents==
The rule for applying an exponent to numbers in a parentheses is best explained with an example.

Here we have 6 to the power 7.

&lt;math&gt;6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 6^{7} = 279936&lt;/math&gt;

Another way to write this is by writing each &lt;math&gt;6&lt;/math&gt; as &lt;math&gt;(2 \cdot 3)&lt;/math&gt;. Just because we wrote &lt;math&gt;6&lt;/math&gt; as &lt;math&gt;(2 \cdot 3)&lt;/math&gt; does not change the final number when multiplying everything in a calculator.

&lt;math&gt;(2 \cdot 3) \cdot (2 \cdot 3) \cdot (2 \cdot 3) \cdot (2 \cdot 3) \cdot (2 \cdot 3) \cdot (2 \cdot 3) \cdot (2 \cdot 3) = 279936&lt;/math&gt;
&lt;math&gt;(2 \cdot 3)^{7} = 279936&lt;/math&gt;

We can change the order and multiply all the &lt;math&gt;2&lt;/math&gt;s in the front and all the &lt;math&gt;3&lt;/math&gt;s in the back and write them as exponents.

&lt;math&gt;2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 279936&lt;/math&gt;
&lt;math&gt; 2^{7} \cdot 3^{7} = 279936&lt;/math&gt;

Since both expressions have to be equal no matter what order the &lt;math&gt;2&lt;/math&gt;s and &lt;math&gt;3&lt;/math&gt;s are multiplied, that means the rule for an exponent on a parentheses is to apply the exponent to all '''multiplied''' numbers in the parentheses.

&lt;math&gt;(2 \cdot 3)^{7} = 279936 = 2^{7} \cdot 3^{7}&lt;/math&gt;

The general rule for an exponent on a parentheses is

&lt;math&gt;(x \cdot y)^{p} = x^{p} \cdot y^{p}&lt;/math&gt;

&lt;math&gt;(a \cdot b \cdot c)^{p} = a^{p} \cdot b^{p} \cdot c^{p}&lt;/math&gt;

===Only Works for Number Multiplied in Parentheses===
This rule only applies when the numbers in the parentheses are '''multiplied''' together. It does not work when numbers are added in the parentheses. Here's an example.

&lt;math&gt;(3 + 5)^{2} = 8^{2} = 64&lt;/math&gt;

&lt;math&gt;\begin{aligned}
(3 + 5)^{2} &amp;= (3 + 5) \cdot (3 + 5) \\
&amp;= 3 \cdot 3 + 3 \cdot 5 + 5 \cdot 3 + 5 \cdot 5 \\
&amp;= 3^{2} + 15 + 15 + 5^{2} \\
&amp;= 64
\end{aligned}&lt;/math&gt;

There is a &lt;math&gt;3^{2}&lt;/math&gt; and a &lt;math&gt;5^{2}&lt;/math&gt; term, but they are not being '''multiplied''' together, and there are two &lt;math&gt;15&lt;/math&gt; terms being added as well.

If we just change the &lt;math&gt;3&lt;/math&gt; and &lt;math&gt;5&lt;/math&gt; in the parentheses to &lt;math&gt;3^{2}&lt;/math&gt; and &lt;math&gt;5^{2}&lt;/math&gt;, the final number is not correct.

&lt;math&gt;(3^{2} + 5^{2}) = (9 + 25) = 34 \neq 64 = 8^{2} = (3 + 5)^{2}&lt;/math&gt;

The rearranging from before needed everything to be '''multiplied''' together, and

&lt;math&gt;(3 + 5)^{2} = 3 \cdot 3 + 3 \cdot 5 + 5 \cdot 3 + 5 \cdot 5&lt;/math&gt;

is not the same as

&lt;math&gt;(3 \cdot 5)^{2} = 3 \cdot 3 \cdot 5 \cdot 5&lt;/math&gt;

'''Rearranging is not possible''' because there is also addition going on, so the rule only applies when everything in the parentheses is multiplied together.

==Practice with Exponents and Parentheses==
You have to simplify any additions in the parentheses first.

The solutions are at the bottom of the page.

==Positive Exponents with Negative Bases==
The same rules apply, but you have to be careful about the final sign.

==Negative Exponents==

==A Zero Exponent with a Non-zero Base==

===Zero to the Zero Power 0^0 is Not Defined===
&lt;math&gt;0^{0}&lt;/math&gt;

==Exponents that are Fractions==
requires first learning about square roots, cube roots, and nth roots

==Practice with Unusual Bases and Powers==
# &lt;math&gt;1^{1}&lt;/math&gt; 
# &lt;math&gt;1^{2}&lt;/math&gt;
# &lt;math&gt;1^{3}&lt;/math&gt;
# &lt;math&gt;1^{100}&lt;/math&gt;

==Solutions to Exercises==
===Practice with Positive Exponents===
# &lt;math&gt;4 \cdot 4 \cdot 4 = 4^{3}&lt;/math&gt;
# &lt;math&gt;5 = 5^{1}&lt;/math&gt;
# &lt;math&gt;7 \cdot 7 \cdot 7 \cdot 7 \cdot 7= 7^{5}&lt;/math&gt;
# 13 to the fourth power &lt;math&gt;= 13^{4}&lt;/math&gt;
# 99 to the power 78 &lt;math&gt;= 99^{78}&lt;/math&gt;
# 9 squared &lt;math&gt;= 9^{2}&lt;/math&gt;
# 57 to the power 8 &lt;math&gt;= 57^{8}&lt;/math&gt;
# 33 cubed &lt;math&gt;= 33^{3}&lt;/math&gt;
# 2 to the 17th power &lt;math&gt;= 2^{17}&lt;/math&gt;

leave off the 1

&lt;math&gt;2^{1} \cdot 2^{1} \cdot 2^{1}&lt;/math&gt;

&lt;math&gt;5^{3} \cdot 5^{0}&lt;/math&gt;

&lt;math&gt;2^{3^{4}}&lt;/math&gt;</text>
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      <comment>Created page with &quot;The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.  Several useful forms of the first law:&quot;</comment>
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      <text bytes="173" sha1="ob3bc6usrbv3vl8c9s1v1vvvzv0syac" xml:space="preserve">The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.

Several useful forms of the first law:</text>
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      <text bytes="683" sha1="6ea0zble651hliyzumdpi22en9ldtcl" xml:space="preserve">The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.

Several useful forms of the first law:

Power formulation:

&lt;nowiki&gt;&lt;math display=&quot;block&quot;&gt;\delta E_T/\delta t = \delta E_{KE} / \delta d_t + \delta E_{PE} / \delta d_t + \delta U / \delta t = \sum_{in} \dot Q_{in}+ \sum_{in} \dot W_{in} + \sum \dot m_{in} ( h_{in} + (1/2) v_{in}^2 +g z_{in}) - \sum \dot m_{out} ( h_{out} + (1/2) v_{out}^2 +g z_{out})   &lt;/math&gt;&lt;/nowiki&gt;

Energy formulation:

&lt;nowiki&gt;&lt;math&gt;\Delta E_{TOT}= \Delta E_{KE} + \Delta E_{PE} + \Delta U = \sum_i Q_i + \sum_j W_j+   m_k ( h_k + (1/2) v_k^2 +g z_k)  &lt;/math&gt;&lt;/nowiki&gt;</text>
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      <text bytes="622" sha1="7z3zyatbb8rfso8fieh8t4b1oirg506" xml:space="preserve">The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.

Several useful forms of the first law:

Power formulation:

&lt;math&gt;delta E_T/\delta t = \delta E_{KE} / \delta d_t + \delta E_{PE} / \delta d_t + \delta U / \delta t = \sum_{in} \dot Q_{in}+ \sum_{in} \dot W_{in} + \sum \dot m_{in} ( h_{in} + (1/2) v_{in}^2 +g z_{in}) - \sum \dot m_{out} ( h_{out} + (1/2) v_{out}^2 +g z_{out})&lt;/math&gt;

Energy formulation:

&lt;math&gt;\Delta E_{Tot} = \Delta E_{KE} + \Delta E_{PE} + \Delta U = \sum_i Q_i + \sum_j W_j + m_k (h_k + (1/2) v_k^2 + g z_k)&lt;/math&gt;</text>
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      <text bytes="638" sha1="dv93oksklje6qj2az5f0g986jq5g50h" xml:space="preserve">The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.

Several useful forms of the first law:

Power formulation:

&lt;math display=&quot;block&quot;&gt;delta E_T/\delta t = \delta E_{KE} / \delta d_t + \delta E_{PE} / \delta d_t + \delta U / \delta t = \sum_{in} \dot Q_{in}+ \sum_{in} \dot W_{in} + \sum \dot m_{in} ( h_{in} + (1/2) v_{in}^2 +g z_{in}) - \sum \dot m_{out} ( h_{out} + (1/2) v_{out}^2 +g z_{out})&lt;/math&gt;

Energy formulation:

&lt;math&gt;\Delta E_{Tot} = \Delta E_{KE} + \Delta E_{PE} + \Delta U = \sum_i Q_i + \sum_j W_j + m_k (h_k + (1/2) v_k^2 + g z_k)&lt;/math&gt;</text>
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      <text bytes="622" sha1="7z3zyatbb8rfso8fieh8t4b1oirg506" xml:space="preserve">The first law of thermodynamics states that energy is neither created nor destroyed, but simply transformed from one form to another.

Several useful forms of the first law:

Power formulation:

&lt;math&gt;delta E_T/\delta t = \delta E_{KE} / \delta d_t + \delta E_{PE} / \delta d_t + \delta U / \delta t = \sum_{in} \dot Q_{in}+ \sum_{in} \dot W_{in} + \sum \dot m_{in} ( h_{in} + (1/2) v_{in}^2 +g z_{in}) - \sum \dot m_{out} ( h_{out} + (1/2) v_{out}^2 +g z_{out})&lt;/math&gt;

Energy formulation:

&lt;math&gt;\Delta E_{Tot} = \Delta E_{KE} + \Delta E_{PE} + \Delta U = \sum_i Q_i + \sum_j W_j + m_k (h_k + (1/2) v_k^2 + g z_k)&lt;/math&gt;</text>
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  <page>
    <title>User:Mitchal Dichter/Pre-University Mathematics/Wishlist/</title>
    <ns>2</ns>
    <id>331791</id>
    <revision>
      <id>2830590</id>
      <timestamp>2026-09-02T20:30:10Z</timestamp>
      <contributor>
        <username>Mitchal Dichter</username>
        <id>2824330</id>
      </contributor>
      <comment>initial creation</comment>
      <origin>2830590</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9" sha1="29dndgnnnfcg5hd558edp661072drgj" xml:space="preserve">blah blah</text>
      <sha1>29dndgnnnfcg5hd558edp661072drgj</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Developing a growth mindset</title>
    <ns>1</ns>
    <id>331792</id>
    <revision>
      <id>2830666</id>
      <timestamp>2026-09-03T04:34:55Z</timestamp>
      <contributor>
        <username>Ckopplemann</username>
        <id>3108346</id>
      </contributor>
      <comment>/* Interest In subject */ new section</comment>
      <origin>2830666</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="460" sha1="flydu6nlvm5dl7p6ci5p9i00gcbbj24" xml:space="preserve">== Interest In subject  ==

I found this topic really interesting and I think the chapter is developing really well. 

I also noticed a slight structuring error in your reference list so I made a small edit to ensure it remained consistent and improved the layout. 

Good luck with the rest of your chapter !!! [[User:Ckopplemann|Ckopplemann]] ([[User talk:Ckopplemann|discuss]] • [[Special:Contributions/Ckopplemann|contribs]]) 04:34, 3 September 2026 (UTC)</text>
      <sha1>flydu6nlvm5dl7p6ci5p9i00gcbbj24</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Sun exposure and protection motivation</title>
    <ns>1</ns>
    <id>331793</id>
    <revision>
      <id>2830677</id>
      <timestamp>2026-09-03T05:19:57Z</timestamp>
      <contributor>
        <username>U3285438</username>
        <id>3103750</id>
      </contributor>
      <comment>/* Food for thought: The health belief model */ new section</comment>
      <origin>2830677</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="841" sha1="q3ri7ebh5kkqvd5oit887ercoth3q88" xml:space="preserve">== Food for thought: The health belief model ==

Hey there, 

Well done on your topic development, I enjoyed reading your book chapter so far!

It is fascinating to see how protection motivation theory (PMT) can be applied to sun-related behaviours. I also noticed your inclusion of the [[wikipedia:Health_belief_model|health belief model]] (HBM) in the 'see also' section, and I wondered if you have considered elaborating on this framework within your book chapter. From my understanding, the HBM suggests cues to action are an important trigger for behaviour change, whereas PMT centres on internal threat appraisal. It might be interesting compare these frameworks in your chapter!

Best regards.   [[User:U3285438|U3285438]] ([[User talk:U3285438|discuss]] • [[Special:Contributions/U3285438|contribs]]) 05:19, 3 September 2026 (UTC)</text>
      <sha1>q3ri7ebh5kkqvd5oit887ercoth3q88</sha1>
    </revision>
  </page>
  <page>
    <title>User:JonAwbrey/Figures and Tables 56</title>
    <ns>2</ns>
    <id>331794</id>
    <revision>
      <id>2830731</id>
      <timestamp>2026-09-03T11:00:35Z</timestamp>
      <contributor>
        <username>JonAwbrey</username>
        <id>29537</id>
      </contributor>
      <comment>add user work page</comment>
      <origin>2830731</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22052" sha1="ogmgzpxgsiyazdapxoo2ystadn5jomn" xml:space="preserve">==Higher Order Sign Relations &amp;bull; Examples==

===Table 36===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 36. Semantics for Higher Order Signs}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object Denoted}&lt;/math&gt;
| &lt;math&gt;\text{Equivalent Signs}&lt;/math&gt;
|-
| width=&quot;50%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| width=&quot;50%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
&amp; = &amp;
\text{“A”}
\\
{}^{\langle} \text{B} {}^{\rangle}
&amp; = &amp;
\text{“B”}
\end{matrix}&lt;/math&gt;
|-
| width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{“A”}
\\
\text{“B”}
\\
\text{“i”}
\\
\text{“u”}
\end{matrix}&lt;/math&gt;
| width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
&amp; = &amp;
{}^{\langle} \text{“A”} {}^{\rangle}
&amp; = &amp;
\text{“}{}^{\langle} \text{A} {}^{\rangle}\text{”}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
&amp; = &amp;
{}^{\langle} \text{“B”} {}^{\rangle}
&amp; = &amp;
\text{“}{}^{\langle} \text{B} {}^{\rangle}\text{”}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
&amp; = &amp;
{}^{\langle} \text{“i”} {}^{\rangle}
&amp; = &amp;
\text{“}{}^{\langle} \text{i} {}^{\rangle}\text{”}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
&amp; = &amp;
{}^{\langle} \text{“u”} {}^{\rangle}
&amp; = &amp;
\text{“}{}^{\langle} \text{u} {}^{\rangle}\text{”}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 37===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 37. Sign Relation Containing a Higher Order Sign}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\ldots
\\[2pt]
\ldots
\\[2pt]
\text{s}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{s}
\\[2pt]
\ldots
\\[2pt]
\text{t}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\ldots
\\[2pt]
\ldots
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 38===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 38. Sign Relation for a Succession of Higher Order Signs (1)}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
x
\\[2pt]
{}^{\langle} x {}^{\rangle}
\\[2pt]
{}^{\langle\langle} x {}^{\rangle\rangle}
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} x {}^{\rangle}
\\[2pt]
{}^{\langle\langle} x {}^{\rangle\rangle}
\\[2pt]
{}^{\langle\langle\langle} x {}^{\rangle\rangle\rangle}
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\ldots
\\[2pt]
\ldots
\\[2pt]
\ldots
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 39===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 39. Sign Relation for a Succession of Higher Order Signs (2)}&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
x
\\[2pt]
s_1
\\[2pt]
s_2
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
s_1
\\[2pt]
s_2
\\[2pt]
s_3
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\ldots
\\[2pt]
\ldots
\\[2pt]
\ldots
\\[2pt]
\ldots
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 40===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 40. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 41===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 41. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 42===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 42. Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 43===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 43. Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 44===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:70%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 44. Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 45===

====PNG====

====LaTeX====

&lt;br&gt;
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:70%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 45. Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;</text>
      <sha1>ogmgzpxgsiyazdapxoo2ystadn5jomn</sha1>
    </revision>
    <revision>
      <id>2830738</id>
      <parentid>2830731</parentid>
      <timestamp>2026-09-03T11:34:43Z</timestamp>
      <contributor>
        <username>JonAwbrey</username>
        <id>29537</id>
      </contributor>
      <comment>reformat tables 40 &amp; 41 as twin tables</comment>
      <origin>2830738</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21170" sha1="7lw7zy5wqm9m7l0dc5h66emw2pligqx" xml:space="preserve">==Work Area==

&lt;br&gt;

{| align=&quot;center&quot; style=&quot;width:80%&quot;
| width=&quot;50%&quot; |
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large; text-align:center; width:90%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 40. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}
| width=&quot;50%&quot; |
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large; text-align:center; width:90%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 41. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}
|}

&lt;br&gt;

===Table 40===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 40. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 41===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 41. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 42===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 42. Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 43===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 43. Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 44===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:70%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 44. Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 45===

====PNG====

====LaTeX====

&lt;br&gt;
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:70%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 45. Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;</text>
      <sha1>7lw7zy5wqm9m7l0dc5h66emw2pligqx</sha1>
    </revision>
    <revision>
      <id>2830739</id>
      <parentid>2830738</parentid>
      <timestamp>2026-09-03T11:45:53Z</timestamp>
      <contributor>
        <username>JonAwbrey</username>
        <id>29537</id>
      </contributor>
      <comment>break out work area</comment>
      <origin>2830739</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21219" sha1="q7zqzltvupz759r60jqcp7gzlsfdu8e" xml:space="preserve">==Work Area==

&lt;br&gt;

{| align=&quot;center&quot; style=&quot;width:80%&quot;
| width=&quot;50%&quot; |
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large; text-align:center; width:90%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 40. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}
| width=&quot;50%&quot; |
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large; text-align:center; width:90%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 41. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}
|}

&lt;br&gt;

==Higher Order Sign Relations &amp;bull; Examples==

===Table 40===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 40. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 41===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 41. Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 42===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 42. Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 43===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:60%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 43. Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 44===

====PNG====

====LaTeX====

&lt;br&gt;

{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:70%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 44. Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{A})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;

===Table 45===

====PNG====

====LaTeX====

&lt;br&gt;
{| align=&quot;center&quot; border=&quot;1&quot; cellspacing=&quot;0&quot; style=&quot;font-size:large;text-align:center;width:70%&quot;
|+ style=&quot;height:30px&quot; | &lt;math&gt;\text{Table 45. Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{B})&lt;/math&gt;
|- style=&quot;height:40px; background:#f0f0ff&quot;
| &lt;math&gt;\text{Object}&lt;/math&gt;
| &lt;math&gt;\text{Sign}&lt;/math&gt;
| &lt;math&gt;\text{Interpretant}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{A} {}^{\rangle} &amp; )
\\
( &amp; \text{A} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{u} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|-
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{B} {}^{\rangle} &amp; )
\\
( &amp; \text{B} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; , &amp; {}^{\langle} \text{i} {}^{\rangle} &amp; )
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
| valign=&quot;bottom&quot; width=&quot;33%&quot; |
&lt;math&gt;\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\end{matrix}&lt;/math&gt;
|}

&lt;br&gt;</text>
      <sha1>q7zqzltvupz759r60jqcp7gzlsfdu8e</sha1>
    </revision>
  </page>
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